Matrix characterization of linear codes with arbitrary Hamming weight hierarchy

The support of an $(n, M, d)$ binary code $C$ over the set $\mathbf{A}=\{0,1\}$ is the set of all coordinate positions $i$, such that at least two codewords have distinct entry in coordinate $i$. The $r$th generalized Hamming weight $d_r(C)$, $1\leq r\leq 1+log_2n+1$, of $C$ is defined as the minimum of the cardinalities of the supports of all subset of $C$ of cardinality $2^{r-1}+1$. The sequence $(d_1(C), d_2(C), \ldots, d_k(C))$ is called the Hamming weight hierarchy (HWH) of $C$. In this paper we obtain HWH for $(2^k-1, 2^k, 2^{k-1}$ binary Hadamard code corresponding to Sylvester Hadamard matrix $H_{2^k}$ and we show that $$d_r=2^{k-r} (2^r -1).$$ Also we compute the HWH of all $(4n-1, 4n, 2n)$ Hadamard code for $2\leq n\leq 8$.

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On cyclic codes in incidence rings

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.43.2006.1.5 ◽

2006 ◽

Vol 43 (1) ◽

pp. 69-77 ◽

Cited By ~ 1

Author(s):

Ricardo Alfaro ◽

Andrei V. Kelarev

Keyword(s):

Directed Graph ◽

Linear Codes ◽

Cyclic Codes ◽

Matrix Ring ◽

Hamming Weight ◽

Quotient Rings

Cyclic codes are defined as ideals in polynomial quotient rings. We are using a matrix ring construction in a similar way to define classes of codes. It is shown that all cyclic and all linear codes can be embedded as ideals in this construction. A formula for the largest Hamming weight of one-sided ideals in incidence rings is given. It is shown that every incidence ring defined by a directed graph always possesses a principal one-sided ideal that achieves the optimum Hamming weight.

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Variance and Covariance of Several Simultaneous Outputs of a Markov Chain

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.1341 ◽

2016 ◽

Vol Vol. 18 no. 3 (Analysis of Algorithms) ◽

Author(s):

Sara Kropf

Keyword(s):

Markov Chain ◽

Hamming Weight ◽

Combinatorial Characterization ◽

Partial Sum ◽

Underlying Graph ◽

Higher Dimensional ◽

Markov Source ◽

Markov Sources ◽

Normally Distributed

The partial sum of the states of a Markov chain or more generally a Markov source is asymptotically normally distributed under suitable conditions. One of these conditions is that the variance is unbounded. A simple combinatorial characterization of Markov sources which satisfy this condition is given in terms of cycles of the underlying graph of the Markov chain. Also Markov sources with higher dimensional alphabets are considered. Furthermore, the case of an unbounded covariance between two coordinates of the Markov source is combinatorically characterized. If the covariance is bounded, then the two coordinates are asymptotically independent. The results are illustrated by several examples, like the number of specific blocks in $0$-$1$-sequences and the Hamming weight of the width-$w$ non-adjacent form.

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Type IV codes over a non-unital ring

Journal of Algebra and Its Applications ◽

10.1142/s0219498822501420 ◽

2021 ◽

pp. 2250142

Author(s):

Adel Alahmadi ◽

Alaa Altassan ◽

Widyan Basaffar ◽

Hatoon Shoaib ◽

Alexis Bonnecaze ◽

...

Keyword(s):

Local Ring ◽

Algebraic Structure ◽

Invariant Theory ◽

Linear Codes ◽

Hamming Weight ◽

Dual Codes ◽

Weight Enumerators ◽

Type Iv ◽

Commutative Local Ring ◽

Torsion Codes

There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: see text] and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

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Characterization of some minimal codes for secret sharing

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500268 ◽

2019 ◽

Vol 12 (02) ◽

pp. 1950026

Author(s):

Sassia Makhlouf ◽

Lemnouar Noui

Keyword(s):

Secret Sharing ◽

Linear Codes ◽

Mds Codes ◽

Secret Sharing Schemes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Dual Codes ◽

Necessary And Sufficient ◽

Minimal Codes

Recently, several authors used linear codes to construct secret sharing schemes. It is known that if each nonzero codeword of a code [Formula: see text] is minimal, then the dual code [Formula: see text] is suitable for secret sharing. To seek such codes Ashikhmin–Barg give a sufficient condition from weights; in [Formula: see text] code [Formula: see text], let [Formula: see text] and [Formula: see text] be the minimum and maximum nonzero weights, respectively. If [Formula: see text] then all nonzero codewords of [Formula: see text] are minimal. In this paper, a necessary and sufficient condition is given for self-dual codes and for MDS codes to verify the inequality (*). Special codes are examined and applied for secret sharing schemes.

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