Weight Distributions Of Linear Codes Research Articles

The weight distribution of codes over finite chain rings

In this work, we determine new linear equations for the weight distribution of linear codes over finite chain rings. The identities are determined by counting the number of some special submatrices of the parity-check matrix of the code. Thanks to these relations we are able to compute the full weight distribution of codes with small Singleton defects, such as MDS, MDR and AMDR codes.

Computing the Partial Weight Distribution of Punctured, Shortened, Precoded Polar Codes

The problem of computing the Hamming weight distribution of linear codes is considered in this paper. A novel method to enumerate all codewords up to a certain Hamming weight for binary linear block codes in general and in particular for the punctured, shortened, precoded polar codes is introduced. The proposed approach performs a recursive decomposition of the codes using construction X4 that is typically used to combine codes of different lengths. This allows to enumerate the low-weight codewords of the overall code as combinations of the low-weight codewords of the component codes. Numerical results show that the proposed approach can efficiently compute the exact partial weight distribution of the 5G New Radio punctured/shortened polar codes with CRC11 and pure polar codes. In the former and latter cases, the low-weight codeword number is up to 106 and 108, respectively. Besides, randomly punctured and shortened polar codes and randomly precoded polar codes are also considered. To the best of the authors’ knowledge, this is the first method able to solve these problems.

A formula on the weight distribution of linear codes with applications to AMDS codes

The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations.

Decoding error probability in the erasure channel and the $\boldsymbol~r$-th support weight distribution

Inspired by a recent work of Shen and Fu (2019), we study the decoding error probability of linear codes over the erasure channel under three decoding principles. We first simplify the explicit formulas obtained by Shen and Fu (2019), showing a much more direct relation to the $r$-th support weight distribution of linear codes. Then we compute the average $r$-th support weight distribution of $[n,k]_q$ linear codes. We resolve an open question of Shen and Fu (2019) concerning the error exponent of the average unsuccessful decoding probability under maximum likelihood decoding for random $[n,nR]_q$ linear codes. We also obtain the error exponent of the average unsuccessful decoding probability for random $[n,nR]_q$ linear codes under list decoding.

Novel blind interleaver parameters estimation based on Hamming weight distribution of linear codes

Interleaving techniques are used to improve the probability of error correction in communication systems. In a non-cooperative context, interleaver parameters must be determined first so that the received data can be decoded into relevant information. This paper proposes blind interleaver parameter estimation method based on the Hamming weight distribution. Conventional methods based on rank distributions suffer from the high computational complexity of Gaussian elimination. In this study, we exploit the fact that the Hamming weight distributions of linear codes differ from those of random sequences owing to the linear dependence of linear codes. By exploiting this property, the proposed algorithm can estimate the interleaver period without a rank calculation. The values of the χ2 test are used to estimate the interleaver period by constructing the Hamming weight distribution that differs the most from the binomial distribution. The experimental results indicate that the proposed algorithm outperforms conventional methods.

The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ l $\end{document}</tex-math></inline-formula> be a prime with <inline-formula><tex-math id="M2">\begin{document}$ l\equiv 3\pmod 4 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ l\ne 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N = l^m $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ m $\end{document}</tex-math></inline-formula> a positive integer, <inline-formula><tex-math id="M6">\begin{document}$ f = \phi(N)/2 $\end{document}</tex-math></inline-formula> the multiplicative order of a prime <inline-formula><tex-math id="M7">\begin{document}$ p $\end{document}</tex-math></inline-formula> modulo <inline-formula><tex-math id="M8">\begin{document}$ N $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ q = p^f $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M10">\begin{document}$ \phi(\cdot) $\end{document}</tex-math></inline-formula> is the Euler-function. Let <inline-formula><tex-math id="M11">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> be a primitive element of a finite field <inline-formula><tex-math id="M12">\begin{document}$ \Bbb F_{q} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ C_0^{(N,q)} = \langle \alpha^N\rangle $\end{document}</tex-math></inline-formula> a cyclic subgroup of the multiplicative group <inline-formula><tex-math id="M14">\begin{document}$ \Bbb F_q^* $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M15">\begin{document}$ C_i^{(N,q)} = \alpha^i\langle \alpha^N\rangle $\end{document}</tex-math></inline-formula> the cosets, <inline-formula><tex-math id="M16">\begin{document}$ i = 0,\ldots, N-1 $\end{document}</tex-math></inline-formula>. In this paper, we use Gaussian sums to obtain the explicit values of <inline-formula><tex-math id="M17">\begin{document}$ \eta_i^{(N, q)} = \sum_{x \in C_i^{(N,q)}}\psi(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M18">\begin{document}$ i = 0,1,\cdots, N-1 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M19">\begin{document}$ \psi $\end{document}</tex-math></inline-formula> is the canonical additive character of <inline-formula><tex-math id="M20">\begin{document}$ \Bbb F_{q} $\end{document}</tex-math></inline-formula>. Moreover, we also compute the explicit values of <inline-formula><tex-math id="M21">\begin{document}$ \eta_i^{(2N, q)} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M22">\begin{document}$ i = 0,1,\cdots, 2N-1 $\end{document}</tex-math></inline-formula>, if <inline-formula><tex-math id="M23">\begin{document}$ q $\end{document}</tex-math></inline-formula> is a power of an odd prime <inline-formula><tex-math id="M24">\begin{document}$ p $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>As an application, we investigate the weight distribution of a <inline-formula><tex-math id="M25">\begin{document}$ p $\end{document}</tex-math></inline-formula>-ary linear code: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathcal{C}_{D} = \{C = ( \operatorname{Tr}_{q/p}(c x_1), \operatorname{Tr}_{q/p}(cx_2),\ldots, \operatorname{Tr}_{q/p}(cx_n)):c\in \Bbb{F}_{q}\}, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where its defining set <inline-formula><tex-math id="M26">\begin{document}$ D $\end{document}</tex-math></inline-formula> is given by <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ D = \{x\in \Bbb{F}_{q}^{*}: \operatorname{Tr}_{q/p}(x^{\frac{q-1}{l^{m}}}) = 0\} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>and <inline-formula><tex-math id="M27">\begin{document}$ \operatorname{Tr}_{q/p} $\end{document}</tex-math></inline-formula> denotes the trace function from <inline-formula><tex-math id="M28">\begin{document}$ \Bbb F_{q} $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M29">\begin{document}$ \Bbb F_p $\end{document}</tex-math></inline-formula>.

Partition-balanced families of codes and asymptotic enumeration in coding theory

We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise asymptotic estimates on the density functions of several classes of codes that are extremal with respect to minimum distance, covering radius, and maximality. The techniques developed in this paper apply to various distance functions, including the Hamming and the rank metric distances. Applications of our results show that, unlike the Fqm-linear MRD codes, the Fq-linear MRD codes are not dense in the family of codes of the same dimension. More precisely, we show that the density of Fq-linear MRD codes in Fqn×m in the set of all matrix codes of the same dimension is asymptotically at most 1/2, both as q→+∞ and as m→+∞. We also prove that MDS and Fqm-linear MRD codes are dense in the family of maximal codes. Although there does not exist a direct analogue of the redundancy bound for the covering radius of Fq-linear rank metric codes, we show that a similar bound is satisfied by a uniformly random matrix code with high probability. In particular, we prove that codes meeting this bound are dense. Finally, we compute the average weight distribution of linear codes in the rank metric, and other parameters that generalize the total weight of a linear code.

Weight Distribution of a Class of Cyclic Codes With Arbitrary Number of Zeros

Cyclic codes have been widely used in digital communication systems and consumer electronics as they have efficient encoding and decoding algorithms. The weight distribution of cyclic codes has been an important topic of study for many years. It is in general hard to determine the weight distribution of linear codes. In this paper, a class of cyclic codes with any number of zeros is described and their weight distributions are determined.

Moments of the support weight distribution of linear codes

In this work, we give the expectation and the covariance formulas for the support weight distributions of linear codes.

The degree of functions and weights in linear codes

Properties of the weight distribution of low-dimensional generalized Reed–Muller codes are used to obtain restrictions on the weight distribution of linear codes over arbitrary fields. These restrictions are used in non-existence proofs for ternary linear code with parameters [74,10,44] [82,6,53] and [96,6,62].

Weight distributions of geometric Goppa codes

The in general hard problem of computing weight distributions of linear codes is considered for the special class of algebraic-geometric codes, defined by Goppa in the early eighties. Known results restrict to codes from elliptic curves. We obtain results for curves of higher genus by expressing the weight distributions in terms of L L -series. The results include general properties of weight distributions, a method to describe and compute weight distributions, and worked out examples for curves of genus two and three.

Weight distributions of linear codes and the Gleason-Pierce theorem

The Gleason-Pierce theorem characterizes those fields for which formally self-dual divisible codes can exist. The ideas underlying the proof of the theorem yield necessary conditions on whether a solution to the MacWilliams identity can be the weight distribution of a linear code. Consequences of this result are an algebraic proof of the non-existence of an [16, 8, 6] f.s.d, binary even code, restrictions on distribution of cosets of codes, and occasional sharpening of upper bounds on the covering radius.

Support weight distribution of linear codes

The main result of the paper is expressions for the support weight distributions of a linear code in terms of the support weight distributions of the dual code.

On the weight distribution of linear codes having dual distance d'&amp;lt;or=k

Using only the principle of inclusion and exclusion, the author derives a formula for the weight distribution of an (n,k) code whose dual code has a minimum distance d'>or=k. The result yields a new condition on the weight distributions of a linear code and its dual which is necessary and sufficient for the code to be a maximum distance separable (MDS) code. Moreover, it shows how the weight distribution for linear MDS codes is obtained in an elementary manner.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The weight distribution of linear codes over GF(q l) having generator matrix over GF(q>)

We study the weight distribution of the linear codes over GF( q l ) which have generator matrices over GF( q) and their dual codes. As an application we find the weight distribution of the irreducible cyclic (23(2 1≈1), 111) codes over GF(2) for all lnot divisible by 11.

Some results on the weight distribution of linear codes

By studying the algebraic structure of the parity check matrix of a linear code we show that the weight distribution is a function only of the quantities N_{\upsilon}^{u} , the number of \upsilon columns of the parity check matrix with rank u . We apply this to obtain a new formula for the weight distribution of the distance 3 s -ary Hamming code and for the distance 4 s -ary conic code. We give the definition of a conic code and some of its properties.