Hamming weight

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Hamming weight is a measure of the number of non-zero elements in a binary string or vector, specifically referring to the count of '1's in the representation. It is commonly used in coding theory and information theory to assess error detection and correction capabilities.

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Key research themes

1. How can new bounds and constructions improve the understanding and application of ternary and higher-weight constant-weight codes in coding theory?

This research theme focuses on the construction, bounds, and optimization of constant-weight codes over larger alphabets such as ternary codes. It aims to improve code parameters like size, minimum distance, and weight, which are critical for error-correction performance and applications in communications and cryptography. The work uses algorithmic approaches such as lexicographic code constructions and greedy algorithms to extend known bounds and create optimal or near-optimal codes. This allows both theoretical advancement and practical code generation for improved coding schemes.

New Bounds for the Maximum Size of Ternary Constant Weight Codes

by Galina Bogdanova

2021

Key finding: Presents new lower bounds for A_3(n,d,w), the maximum size of ternary constant-weight codes, by employing lexicographic codes constructed with greedy algorithms. Specifically, codes of length n from 5 to 10 and minimum... Read more

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Computer construction of quasi-twisted two- weight codes

by eric chen

2023

Key finding: Demonstrates a generalization that consta-cyclic codes of composite length can be expressed as quasi-twisted codes, enabling the computer-based construction of numerous quasi-twisted two-weight codes. The paper identifies... Read more

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Generalized Hamming weights of three classes of linear codes

by Hongfeng Wu

2023, Finite Fields and Their Applications

Key finding: Investigates generalized Hamming weights (GHWs) of three classes of linear codes, constructed through defining sets, and determines the GHWs for several cases, particularly solving an open problem in the semiprimitive case... Read more

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2. What novel algebraic and combinatorial methods allow computation and bounding of generalized Hamming weights in binary and cyclic codes?

This research area investigates methods to compute or tightly bound the generalized Hamming weights (GHWs) of binary linear codes, including BCH codes and their duals, which reflect code performance beyond minimum distances and have applications in error detection/correction and cryptography. It includes the use of algebraic geometry tools such as graded free resolutions, binomial ideals, and Gröbner bases, as well as combinatorial structures of codeword supports and minimal codewords. Such methods aim to overcome computational challenges to determine GHWs more efficiently and understand their implications.

Free Resolutions and Generalized Hamming Weights of Binary Linear Codes

by Edgar Martinez-Moro

2023, Mathematics

Key finding: Introduces an approach linking the computation of the first and second generalized Hamming weights of binary linear codes to graded free resolutions of monomial ideals associated with subsets of codewords, notably a Gröbner... Read more

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Generalized Hamming weights of linear codes

by Torleiv Klove

2016, IEEE Transactions on Information Theory

Key finding: Develops foundational properties and theoretical machinery for generalized Hamming weights of linear codes, including BCH codes. Notably, shows for double-error-correcting BCH(2^m,5) codes that d_2=8 and extends to higher... Read more

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Generalized Hamming weights of three classes of linear codes

by Hongfeng Wu

2023, Finite Fields and Their Applications

Key finding: Employs additive character sums and Gauss periods to establish explicit expressions and partial determinations for generalized Hamming weights of three classes of linear codes defined by special subsets (defining sets). The... Read more

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3. How can geometric and combinational properties of subsets of the Hamming cube inform bounds on distance matrices and related metric properties?

This line of research studies finite metric spaces formed by subsets of the Hamming cube, focusing particularly on the relationship between their distance matrices and underlying geometric structures like affine independence. It explores how classical results on trees and distance matrices extend to general subsets, and uses concepts such as negative type metrics and S-embeddings to establish bounds on quantities related to inverses of distance matrices, with implications for combinatorial optimization and metric geometry.

A problem on distance matrices of subsets of the Hamming cube

by Ian Doust

2025, Proceedings of the American Mathematical Society, Series B

Key finding: Proves that for n+1 affinely independent points in the Hamming cube H_n, the sum of all entries in the inverse of their distance matrix is at least 2/n, confirming a conjecture that the minimum is attained by unweighted... Read more

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All papers in Hamming weight

Generalization of the Norse bounds to codes of higher strength

by Patrick Sole

1991, IEEE Transactions on Information Theory

The Norse bounds state that all codes of strength 1 and length IZ have covering radius at most n /2 and all self-complementary codes of strength 2 and length n have covering radius at most ( n -&)/2. We generalize this to arbitrary even... more

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On the Coding Channel Capacity and Cutoff Rate of Frequency-hopping communications with non-ideal interleaving

by Shaul Laufer

The Sixteenth Conference of Electrical and Electronics Engineers in Israel,

Frequency-hopping communications over channels with partial band noise jamming, both with and without side information, is considered. The capacity and the cutoff rate are calculated both for the case of ideal interleaving and the case of... more

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A new approach to the general minimum distance decoding problem: The zero-neighbors algorithm

by Lev Levitin

1985, IEEE Transactions on Information Theory

Absfracf-Minimum distance decoding (MDD) for a general error-correcting linear code is a hard computational problem that recently has been shown to be N&hard. The complexity of known decoding algorithms is determined by min(2k,2"-k),... more

of |N>| have been estimated by use of (A7). The results show that using ZNA substantially reduces complexity, although for finite n the gain is smaller than that in the asymptotical case. It is interesting to compare the number of zero neighbors with the number of codewords and coset leaders for finite values of n and k. These numbers for some typical linear error-correcting codes are presented in Table I. The values

(M = 2* = 2%); Method 3 and Method 4 in Section I (M = 2”-* = 2”0-®)): and decoding by the ZNA (M = |N,| for the space complexity where |N,| is given by (A8) for R > R*).

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Residual error rate of binary linear block codes (Corresp.)

by Gerard Cohen

1976, IEEE Transactions on Information Theory

Transposing a computation of Mac Williams we derive e30 an exact expression for, and a straightforward upper bound on, the residual error rate of a binary block code with a "standard" decoder. We also give series expansions for decoding... more

Fig. 2. Sequential code reduction circuit utilizing idempotent.

Series expansions of PyN-: The computations are the same as in Section II. One has only to change, in the formula giving Pg (or Pea = Pa — AoPca), Aj to iA;. Thus, by (3), we readily obtain R,e is the average number of errors per information bit after decoding. The code is assumed to be systematic with d = 2t + 1. For the complete decoder shown in Fig. 1, we now determine R,o. Gate G1 is in position “a” when D; is successful (event of prob- ability Pz); then, under the H hypothesis, R,e equals N, e/n. When D; fails (an event of probability 1 — Pa), G1 comes into position “b” and Rre equals N.’/n, where N,’ is the average weight of the error vector e, given that D; fails. The number of words of weight Rk, not decodable by D,, i.e., located at distance greater than t from every codeword, is

fig. 1. Simplified sequential code reduction circuit for (31, 16) code.

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Complete Distances of All Negacyclic Codes of Length $2^{s}$ Over $\BBZ _{2^{a}}$

by Hai Dinh

2007, IEEE Transactions on Information Theory

Various kinds of distances of all negacyclic codes of length 2 s over 2 are completely determined. Using our structure theorems of negacyclic codes of length 2 s over 2 , we first calculate the Hamming distances of all such negacyclic... more

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Negacyclic Codes of Length<tex>$2^s$</tex>Over Galois Rings

by Hai Dinh

2005, IEEE Transactions on Information Theory

Codes over the ring of integers modulo 4 have been studied by many researchers. Negacyclic codes such that the length of the code is odd have been characterized over the alphabet 4 , and furthermore, have been generalized to the case of... more

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ZOT-Binary: A new

by Shahram Jahani

2013

In this paper we present a new numbering system with an efficient application on Big-Integer multiplication. The paper starts with an introduction to a new redundant positional numbering system known as “Big-Digit Numbering System ”... more

Figure 1: Nonzero Big-Digits distribution in ZOT-Binary representation based on 10,000 random bit In general, suppose the length of a Big-Digit is n and BD-MLUT support only Big-Digits less than m, then the following conversions can be applied. n The memory requirement to store the MLUT for n ZOT-CM is only 25 byte as shown below:

in the readings. Experiment was conducted or AMD Phenom (TM), 9950, Quad core processor 2.6 GHz, 3.25GB RAM, with Windows XF (Service Pack 2) Professional and Dev-C++ versior 4.9.9.2 compiler. Table 8. Execution Time Ratio Of ZOT-CM Multiplication Algorithm Against Classical And Karatsuba Multiplication Algorithms

Table 3. Big-One-Big-One MLUT (BOBO-MLUT) Table 4. Big-Two-Big-Two MLUT (BTBT-MLUT)

Table 2. Summary Of Contradictions For Two Different ZOT-Binary Representations In Table 2, we listed all valid cases of c, and b, (symmetrical cases have been ignored) and we showed that for each case, there is a contradiction. Therefore, by contradiction, the ZOT-Binary represents non-negative integers uniquely. Definition 5 introduces the ZOT-Binary representation. The representation is named ZOT- Binary to highlight the base, which is base two, and the digits used in the representation, which are Zero, Big-One and Big-Two.

Step 1: Initializing: Set all c;, to zero. Let A = (py ...,@2,Q4,Q9)2 be a binary number and let the ZOT-Binary representation of A be (Cy ++) C2,Cq,Co)2, Where c; = (Ci, Cj) represents the i-th Big-Digit in A. c;, denotes the type of c; and Cj, denotes the length of c;. Following are the steps to convert a binary number to its ZOT-Binary representation. Step 2: Identifying the position of nonzero in BD: Scan the binary number from right to left to identify the first nonzero bit a;. This indicates the beginning of a new nonzero in BD.

Table 6. Distribution of nonzero Big-Digits in ZOT-Binary representation in range of 128 bits to 32 kbits Table 6 shows another perspective of the Big- Digits distribution in range of 128 bits to 32 kbits random integer. It indicates that ZOT-Binary representation reduce the percentage of nonzero symbols to 21.86%, while the average of Hamming weight for MOF and NAF is 50% [4] and 33% [21] respectively.

Table 5. Big-One-Big-Two MLUT (BOBT-MLUT)

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On the weight distribution of linear block codes formed from convolutional codes

by Andrew Viterbi

1996, IEEE Transactions on Communications

In this note we explain how to obtain the weight enumerator and the performance of linear block codes formed in several distinct ways from a convolutional code. SSUME a rate 1/2 binary convolutional code of con-A straint length (m+ 1).... more

Fig. 3. Sixteen-state, binary, rate 1/2 convolutional encoder.

Fig. 2. Terms of [A*] which contribute to the weight enumerator (a) ZT, (b) generalized ZT with m’ = 2, (c) direct truncation, (d) TB, and (e) generalized TB with m’ = 2.

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Differential Fault Analysis on the Families of SIMON and SPECK Ciphers

by Debdeep Mukhopadhyay

2014, 2014 Workshop on Fault Diagnosis and Tolerance in Cryptography

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Generalized Hamming weights for linear codes

by ESTEBAN ESCOBAR ORTIZ

1991, IEEE Transactions on Information Theory

Error control codes are widely used to increase the reliability of transmission of information over various forms of communications channels. The Hamming weight of a codeword is the number of nonzero entries in the word; the weights of... more

The Maple source code for these follows in the next subsection. Appendix 7.2 Maple Source Code

Let H be the parity-check matrix Proof: We consider the cases m > 5 and odd first. By the Griesmer bound d3(C’) > 10 for any code with d = 5 and k > 3, so it is enough to find a D C C, with dim(D) = 3, with |x(D)| = 10. We will use codewords of weight 5 to construct such a D. The first step is to study words of weight 5 with 1’s in two given positions. Let H be the parity-check matrix We fix 3! = s and 34 =t and do a change of variables

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Towards security limits in side-channel attacks

by Jean-jacques Quisquater

2006

In this paper, we consider a recently introduced framework that investigates physically observable implementations from a theoretical point of view. The model allows quantifying the effect of practically relevant leakage functions with a... more

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Algebraic immunity for cryptographically significant Boolean functions: analysis and construction

by Deepak Kumar

2006, Information Theory, IEEE …

Recently, algebraic attacks have received a lot of attention in the cryptographic literature. It has been observed that a Boolean function used as a cryptographic primitive, and interpreted as a multivariate polynomial over 2 , should not... more

A drawback of the function @, is that it is unbalanced. This happens since ¢2 = %1% is unbalanced. If one starts the con- struction with @ as affine function, then the function oz will always be balanced as 4; = 21 +22+(i+j7) mod 2 for? > 0, 4S 70 Now we nrecent cqme ohcervatinne in thic reoard

the point of coordinates (7,27) and arriving for the first time to a point of coordinates (0,27) (where 2/ can be any integer between 0 and 2(j —7)), and constituted of the concatenation of elementary paths going from a point of coordinates (s, 2) to a point of coordinates (s + «. 24 — 2).c € £-1.0.1%. equals 1 for! = k — 2; it equals 0 for] = & — 3; and for 1 < k —4, it equals the number of paths from the point of coordinates (7,27) where 2 = 2; 7 = k — 2 to the point of coordinates (7,27) where i = 2; 7 = 1+ 2 which do not cut the axis of equation z = 0 (indeed, the two last elementary paths are necessarily (2, 27+ 4) — (1, 22+ 2) and (1, 27+ 2) — (0, 21)). Note that 7 is null if /: — 7 is odd. We assume now that 4; — / is even. Then 7 equals the number of all paths between these two points (the points (2, 2k — 4) and (2, 2/ + 4)), that is, (2175), minus the number of paths cutting the axis z = 0. This last number equals the number of paths from the point of coordinates (2,27) where i = 2; j = k—2 to the point of coordinates (7, 27) where (¢ = —2; 7 = 1+ 2) (replacing the lower part of each path cutting the axis by its mirror image with respect to this axis). We have then

optimal but poor nonlinearity. The results are same for both the cases so we do not write them separately in brackets.

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Evaluation of Slovene local pig breed Krškopolje for carcass and meat quality

by G. Fazarinc and

2003, Czech J. Anim. Sci

... be related to lit-erature data since ham processing losses are lower in heavier and/or fatter hams (review of Russo and Nanni Costa, 1995). ... Fujii J., Otsu K., Zorzato F., De Leon S., Khanna VK, Weiler JE, O&#x27;Brien PJ,... more

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Fault-Based Side-Channel Cryptanalysis Tolerant Rijndael Symmetric Block Cipher Architecture

by Piyush Mishra

2001

Fault-based side channel cryptanalysis is very effective against symmetric and asymmetric encryption algorithms. Although straightforward hardware and time redundancy based Concurrent Error Detection (CED) architectures can be used to... more

Table 2: FPGA implementation of 128-bit Rijndael CED architectures 4. References

Figure 4: Rijndael encryption with operation-level CED

Table 1: Comparison of 128-bit Rijndael CED architectures

A closer look at Rijndael reveals that the inverse relationship between encryption and decryption exists at the round level as well. Passing the input data successively through one encryption round and the corresponding decryption round yields the original data. For Rijndael, the firs round of encryption corresponds to the tenth round of decryption; the second round of encryption corresponds to the ninth round of decryption and so on. Based on this observation s the computations can be checked at the round level. At the beginning of each Rijndae encryption round corresponding round key and the input data is stored in registers before feeding to the round module. After one round of encryption, the output is fed to the corresponding round of Rijndael decryption. The output of decryption round is calculated using the stored key and then compared with the input data that was previously saved. If there is a mismatch, Rijndae encryption is halted and an error is signaled. Rijndael encryption with round-level CED is shown in Figure 3. in Figure 3. encryption is halted and an error is signaled. Rijndael encryption with round-level CED is shown

Algorithm-level CED approach shown in Figure 2 exploits the inverse relationship at the algorithm-level. Plain text is processed through the encryption module, which is then disabled (or processes next block of data) and the decryption module is enabled to decrypt the cipher text. The output of decryption is compared with the input plain text. An error signal is set and the faulty cipher text is suppressed when there is a mismatch. Algorithm-level CED during decryption is similar. The area overhead includes an additional register to store the original input, a comparator, and a few multiplexers at the input of the encryption, decryption and comparator modules. Since the round keys are stored in a key RAM, next block of plain text (cipher text) cannot be processed until both encryption and decryption of current plain text (cipher text) is finished (since otherwise different round keys need to be accessed simultaneously by the encryption and decryption modules). Hence, the time overhead of encryption for one block of data with algorithm-level CED is the time it takes for encryption (without CED) of one block of data. The time to encrypt N blocks ot input data using algorithm-level CED is 2NU the time for basic encryption of one block of data. The latency between the deliberate introduction of a fault and its detection may be sufficient tc obtain other useful side channel information. Although the time overhead of this approach is identical to the time overhead of encryption (decryption) with time redundancy based CED, it can detect permanent faults as well. If the keys are stored in a register file or are stored in twc different key RAMs - encryption key RAM and decryption key RAM, encryption of the current block of data can be carried out concurrently with the decryption (for CED) of the previous block of data. This reduces the total time for encrypting N blocks of data to (N+1) U the time fot basic encryption of one block of data.

Se MAS ee Pe See EN ee a ee A 128-bit block, 128-bit key Rijndael encryption supports 10 rounds, each round using | round key [20]. An additional key is used in pre-processing. Rijndael operates on a two-dimensional table of plain text bytes called the state. Operations used in a round of Rijndael are: a non-linear byte substitution operation (S-BOX), a cyclic left shift of the rows in the state (shift row), GF (2° multiplication with a constant of every column of the state (mix column) and exclusive-or of round key with the state (key-xor). It has an iterative looping structure as shown in Figure 1. After pre processing, the input plain (cipher) text is subjected to one round of encryption (decryption) where a series of operations are performed on it and the round key(s) to generate the intermediate cipher text. The intermediate cipher text is then used as input to the next round. We developed VHDL' models and FPGA implementation for Rijndael encryption [16]. ' Very High Speed Integrated Circuits Hardware Description Language that is widely used in the industry.

Figure 5: Interconnections between s-box and s-box” operations FPGA implementation Such an operation-level CED further improves the latency of detecting a fault that has been deliberately introduced by an attacker thereby minimizing the amount of side channel information that is revealed. In addition, it localizes the fault to the hardware implementing the operation. The design with operation-level CED is complex and has additional performance penalty due to additional multiplexers in the critical path. Figure 5 shows interconnections between the S-BOX operation of the encryption round and the S-BOX" operation of the decryption round. We implemented the 128-bit Rijndael private key encryption algorithm with different CED approaches in Xilinx Virtex XCV1000BG560-6 FPGA. We modeled Rijndael and all CED architectures in VHDL, verified the functionality using Modelsim VHDL simulator and synthesized the data paths using Synplify Synplicity synthesis tool and Xilinx place and route tool. One round of Rijndael encryption (decryption) consumes four clock cycles. Therefore, encrypting (decrypting) one 128-bit block of plain (cipher) text consumes 11 rounds x 4 = 44 clock cycles. The round keys are stored in a key RAM for use during both encryption and decryption.

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Constructions of binary constant-weight cyclic codes and cyclically permutable codes

by Quy Nguyen

1992, IEEE Transactions on Information Theory

A general theorem is proved showing how to obtain a constant-weight binary cyclic code from a p-ary linear cyclic code, where p is a prime, by using a representation of G F ( p ) as cyclic shifts of a binary p-tuple. Based on this... more

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Generalized Hamming weights of three classes of linear codes

by Hongfeng Wu

2017, Finite Fields and Their Applications

The generalized Hamming weights of a linear code have been extensively studied since Wei first use them to characterize the cryptography performance of a linear code over the wire-tap channel of type II. In this paper, we investigate the... more

Remark 4 Similar corollaries can be obtained using the method in the above subsec- tion, so we omit here.

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Error-Correction Capability of Binary Linear Codes

by Torleiv Klove

2005, IEEE Transactions on Information Theory

The monotone structure of correctable and uncorrectable errors given by the complete decoding for a binary linear code is investigated. New bounds on the error-correction capability of linear codes beyond half the minimum distance are... more

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The Higher Weight Enumerators of the Doubly-Even, Self-Dual [48, 24, 12] Code

by Thomas Britz and

2007, IEEE transactions on …

Fig. 2. The fourth higher weight enumerators of G and U. nonzero codeword weight of C' is d > 1 and X,Y C E are dis- joint sets of coordinates with |X| + |Y| < d— 1, then each element of C'/X\Y is simple. By (6), each of the first ¢d — 1 recursion levels involves a sum of the form Ton. (x, y) + Tai /e(2, y). The task of cal- culating Tc(2x, y) may therefore be split up into 27? subtasks, each to calculate the Tutte polynomial of a linear code of the form C'/X\Y. The sum of these 2¢~? polynomials is To (x,y).

Fig. 4. The fifth higher weight enumerators of G and U. yhere d+ is the minimum nonzero weight of the dual code (C’)+.

Fig. 1. The fourth higher weight enumerators of RM (2, 5), QR, and F.

Fig. 3. The fifth higher weight enumerators of RM (2,5), QR, and F.

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A new integrated approach to the design of low-complexity FIR filters

by Chip Hong Chang

2006, 2006 IEEE International Symposium on Circuits and Systems

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A Family of Two-Weight Irreducible Cyclic Codes

by Nimalsiri Pinnawala

2010, Ieee Transactions on Information Theory

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Type IV self-dual codes over rings

by Patrick Sole and

1999, Information Theory, …

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On computing the Hamming distance

by Gerzson Keri

2004, Acta Cybernetica

Methods for the fast computation of Hamming distance developed for the case of large number of pairs of words are presented and discussed in the paper. The connection of this subject to some questions about intersecting sets and Hadamard... more

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On the computation of the minimum distance of low-density parity-check codes

by Marc Fossorier

2004

Low-density parity-check (LDPC) codes in their broader-sense definition are linear codes whose paritycheck matrices have fewer 1s than 0s. Finding their minimum distance is therefore in general an NP-hard problem; in other words there... more

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Rate Variable, Multi-Binary Turbo Codes with Controlled Error-Floor

by Sandro Bellini

2000, IEEE Transactions on Communications

In this letter we propose rate variable turbo codes based on the parallel concatenation of tailbiting Recursive Systematic multi-binary (m-ary) convolutional codes. Rate variability is not achieved by puncturing, which can have adverse... more

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Performance Evaluation of Parallel Concatenated Codes

by Guido Montorsi

1995, … , 1995. ICC'95 Seattle,'Gateway to …

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Lecture notes in computer science

by Radwanullah Kakar

1996

Since 1984, the Artificial Intelligence: Methodology, Systems, and Applications (AIMSA) conference series has provided a biennial forum for the presentation of artificial intelligence research and development. The conference covers the... more

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On the covering radius of binary (14, 6) codes containing the all-one vector

by Krassimir N Manev

1988, IEEE Transactions on Information Theory

It is shown that the covering radius of any binary linear [14,6] code containing the all-one vector is at least 4. Since the minimum covering radius of a binary linear [14,6] code is 3, this shows that in general the minimum of the... more

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On Type IV self-dual codes over

by Stefka Bouyuklieva

2002, Discrete Mathematics

Recently, Type IV self-dual codes over rings of order 4 have been introduced as self-dual codes over the rings with the property that all Hamming weights are even. All Type IV self-dual codes over Z4 of lengths up to 16 are known. In this... more

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Analysis and Simulation of Turbo Codes Performance Over Rice Fading Channels

by Guido Montorsi

ieeexplore.ieee.org

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Contextual Ontologies

by Faiez Gargouri

2006

Contextual ontologies are ontologies that characterize a concept by a set of properties that vary according to context. Contextual ontologies are now crucial for users who intend to exchange information in a domain. Existing ontology... more

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Generalized Hamming weights of linear codes

by Torleiv Klove

1992, IEEE Transactions on Information Theory

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Generalizations of Gleason's theorem on weight enumerators of self-dual codes

by Neil Sloane

1972, IEEE Transactions on Information Theory

Gleason has recently shown that the weight enumerators of binary and ternary self-dual codes are polynomials in two given polynomials. In this paper it is shown that classical invariant theory permits a straightforward and systematic... more

- Then there are 7 basic invariants, of degrees 4,6,8,8,10, 12,14, as follows: ER een Nee Vee OORT A ee TR eT , a ae 5.3.3. Hamming Weight Enumerators over GF(q): In going from formally self-dual enumerators to self-dual codes, for general g nothing can be added to what was said in (5.2.3). But for small g we can study the effect of imposing the restriction that the Hamming weights of all codewords be divisible by a constant. According to the following theorem there are four cases in which this can happen. la ft Dl Ti ot DE: i. 7. ees ol ee be, so, rr cs, ©.) These invariants are unchanged if B is replaced by iB. The group generated by A, iB, and C has order 336 and is im- portant in elliptic function theory and in geometry. (Maschke [23b], Edge [12].) A set of basic invariants was given by Maschke in 1893. For completeness and to correct some errors in Maschke’s work (and also because of the inaccessibility of [23b]), we reproduce them here. Let the exponents of ¢,,x, y,z count the occurrences of 0,+1,+2,+3 in a codeword; and write ¢, instead of 2x, t; instead of 2y, and f, instead of 2z. Po wee + >

The following examples of complete weight enumerators of codes over GF(3) show that all four polynomials are necessary. The second code is self-dual, the others equivalent to their duals. 4.6. Invariants of Large Groups and as basic polynomials we may take a*,b?,f*, and abf?*, with the syzygy (abf?)? = a? +b? - f*. If the invariants of a subgroup H are known, then the invariants of G may be found from the following theorem.

is a polynomial in A,C,B?,BD*,D*. The following examples of self-dual codes show that all five polynomials are neces- sary. For generators of these codes see Pless’s list [29]. For each weight enumerator, we describe the group G of transformations under which it is invariant by giving the generators (enclosed within diamond brackets) and the degree n (the number of variables on which G acts). Then we give the Molien series (A) (4.2.8), and where possible a set of basic polynomial invariants f,,°*+,f5915°''>Gm» the g; being indicated by asterisks. Any weight enumerator is a sum of terms f,"',---,f,'"9*, j,€Z*, e=0orlis< ism.

Let # = Wand B = B* be self-dual codes. It follows from (3.4.1)-(3.4.4) that their joint weight enumerator F «t.a(a,b,c,d) is invariant under the transformation If «,Z are binary self-dual codes, then ¥,, g is invariant under some additional transformations. Let u € %/, ve &. Then n = i(u,v) + j(u,v) + k(u,v) + (u,v), wt @) = k(u,v) + (u,v), and wt (v) = j(u,v) + [(u,v) must all be even. There- fore i,j,k,/ are either all odd or all even, and so $.4 % is invariant under the transformation (3.7.1) for the following values of 7:

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Verification of Analog and Mixed-Signal Circuits Using Hybrid System Techniques

by Thao Dang

2004

In this paper we demonstrate a potential extension of formal verification methodology in order to deal with time-domain properties of analog and mixed-signal circuits whose dynamic behavior is described by differential algebraic... more

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Self-adaptive Differential Evolution

by Ayed Salman

2005

Differential Evolution (DE) is generally considered as a reliable, accurate, robust and fast optimization technique. DE has been successfully applied to solve a wide range of numerical optimization problems. However, the user is required... more

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On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings

by Steve Szabo

2009, Advances in Mathematics of Communications

Repeated root Cyclic and Negacyclic codes over Galois rings have been studied much less than their simple root counterparts. This situation is beginning to change. For example, repeated root codes of length p s , where p is the... more

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On some linear codes that are proper for error detection

by Torleiv Klove

2012, 2012 Information Theory and Applications Workshop

It is shown that, for all prime powers q and all k ≥ 3, if n ≥ (k − 1)q k − 2 q k −q q−1 , then there exists an [n, k; q] code that is proper for error detection.

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A Linear Construction for Certain Kerdock \\ and Preparata Codes

by Patrick Sole

1993, Bulletin of the American Mathematical Society

The Nordstrom-Robinson, Kerdock, and (slightly modified) Preparata codes are shown to be linear over Z4 , the integers mod 4 . The Kerdock and Preparata codes are duals over Z4 , and the Nordstrom-Robinson code is self-dual. All these... more

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Type IV-II codes over Z4 constructed from generalized bent functions

by Sanja Rukavina

2021, arXiv (Cornell University)

A Type IV-II Z 4-code is a self-dual code over Z 4 with the property that all Euclidean weights are divisible by eight and all codewords have even Hamming weight. In this paper we use generalized bent functions for a construction of... more

Remark 4.2. A 1-(32,8,7) design marked with * in Table 2 is a (4,7; 2)-net, i.e., an affine resolvable 1-design. Its block intersection graph Go is a strongly regular graph with parameters (28, 15,6, 10).

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Winner-Imposing Strategyproof Mechanisms for Multiple Facility Location Games

by angelina zhou

2010

We study Facility Location games, where a number of facilities are placed in a metric space based on locations reported by strategic agents. A mechanism maps the agents’ locations to a set of facilities. The agents seek to minimize their... more

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Computational capabilities of feedforward neural networks: the role of the output function

by Jose Subirats

Proceedings of the XII CAEPIA

We present an analysis of the computational capabilities of feed-forward neural networks focusing on the role of the output function. The space of configurations that implement a given target function is analyzed for small size networks... more

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Generalized Hamming weights of q-ary Reed-Muller codes

by P. Heijnen and

1998, IEEE Transactions on Information Theory

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the q-ary Reed-Muller codes. For the latter codes it is shown that this... more

Example 4.17 Let R = F3[X,, X2, X3]. Consider the graded lexicographic order on the monomials in the variables X,, X2, X3. Let f; be the ith mono- mial in this order. and so on. Let P be the set of all 27 distinct points in F3. Let » be the evaluation map from R to F3’ defined by y(f) = (f(Pi),---, f(P27)). The points P,,...,P.7 ordered in the lexicographic order are tabulated below

Example 6.10 Let gq = 4, r=8 and u=v=m==3. Then the tree looks like: The pairs (u;,m,;), the sets N(u;,m;) and the @; are given in the following

In this enumeration those points for which the sum of their entries is greater than 3 are printed boldface. Their labels in the lexicographic order gives us the generalized weights of C(20). Example 5.13 Consider the code C(20) = RMs3(2,3). This is a continua tion of Example 4.18. Enumerate all the points P,,..., P27 of F3 in lexico graphic order.

The pairs (u;,m;) and sets N(u;,m,;) are given in the following table.

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Properties of Finite Response Input Sequences of Recursive Convolutional Codes

by han thien thien

1999, Lecture Notes in Computer Science

A recursive convolutional encoder can be regarded as an innite impulse response system over the Galois Field of order 2. First, in this paper, we i n troduce nite response input sequences for recursive convolutional codes that give nite... more

Table 1. Basic FRISs for M = 2 Q(D) =1+D+D?". In this section, we will give a list of basic FRISs for recursive convolutional encoders with M=2, 3, 4 and 5. The following basic FRISs have been obtained from an exhaustive search since there is no known method to find them.

Fig. 1. The structure of a recursive systematic convolutional encoder of rate 1/2.

Table 2. Basic FRISs for M = 3 Q(D) =14+ D+ D*.

Table 3. Basic FRISs for M = 4 Q(D) =1+D+D%.

Fig. 2. The state diagram for a primitive polynomial Q(D) = 1+ D+ D®. 3 Properties of Finite Response Input Sequences (FRIS. We have the following theorems about F(D).

Fig. 3. The structure of a turbo encoder of rate 1/3.

Table 4. Basic FRISs for M = 5 Q(D) =1+D+D°?4+ D?4+D*. 5 Exemples of Application

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The effect of ractopamine hydrochloride (Paylean®) on lean carcass yields and pork quality characteristics of heavy pigs fed normal and amino acid fortified diets

by F. McKeith

2009, Meat Science

Effects of ractopamine hydrochloride (RAC) on carcass parameters in heavy weight (133.24 ± 8.07 kg) finishing pigs (n = 278) given amino acid fortified (AA) or 16% crude protein (CP) diets were evaluated. A total of seven experimental... more

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Weight enumerators and gray maps of linear codes over rings

by Bahattin Yildiz

2006

The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the... more

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Chain Codes and Spherical Tits Buildings

by Marcelo Firer

2015

To an n-dimensional vector space V over a finite field Fq there is an (naturally) associated spherical building of type An−1. The chambers of such a building are maximal flags: maximal sequences of nested sub-spaces of V. In the case q =... more

Figure 1: Geometric realization of A9(2) building.

is a disjoint union. The result follow of the Principle of Inclusion-Exclusion:

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Generalized Bose–Lin Codes, a Class of Codes Detecting Asymmetric Errors

by Torleiv Klove

2000, IEEE Transactions on Information Theory

Bose and Lin introduced a class of systematic codes for detection of binary asymmetric errors. In this note, we describe a generalization to q-ary asymmetric error detecting codes. For these codes, the possible undetectable errors are... more

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New constructions of optimal cyclically permutable constant weight codes

by Oscar Moreno

1995, IEEE Transactions on Information Theory

Three new constructions for families of cyclic con stant weight codes are presented. All are asymptotically optimum in the sense that in each case, as the length of the sequences within the family approaches infinity, tbe ratio of family... more

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Extended cubes

by Jennifer Seberry and

2011, Proceedings of the 6th ACM Symposium on Information, Computer and Communications Security - ASIACCS '11

In this paper, we propose an efficient method for extracting simple low-degree equations (e.g. quadratic ones) in addition to the linear ones, obtainable from the original cube attack by Dinur and Shamir at EUROCRYPT 2009. This extended... more

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An interleaver design algorithm based on a cost matrix for turbo codes

by han thien thien

2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060)

In this paper the design of interleavers for Turbo Codes is considered. The proposed algorithm is based on a Hamming weight cost matrix. It optimizes both the minimal distance of Turbo Codes and the passing of extrinsic information.... more

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