On the Minimum Distances of Non-Binary Cyclic Codes

A simple method based on Newton's identities and their extensions is presented for determining the actual minimum distance of cyclic codes. More significantly, it is shown that this method also provides a mechanism for generating the type of syndrome matrices needed by Feng and Tzeng's (see ibid., vol.40, p.1364-1374, Sept. 1994) new procedure for decoding cyclic and BCH codes up to their actual minimum distance. Two procedures for generating such matrices are given. With these procedures, we have generated syndrome matrices having only one class of conjugate syndromes on the minor diagonal for all binary cyclic codes of length n<63 and many codes of length 63/spl les/n/spl les/99. A listing of such syndrome matrices for selected codes of length n<63 is included. An interesting connection of the method presented to the shifting technique of van Lint (1986) and Wilson is also noted.

In this paper, an exact distance spectrum formula for the largest minimum Hamming distance of finite-length binary block codes is presented. The exact formula indicates that the largest minimum distance of finite-length block codes can be fully characterized by the information spectrum of the Hamming distance between two independent and identically distributed (i.i.d.) random codewords. The distance property of finite-length block codes is then connected to the distance spectrum. A side result of this work is a new lower bound to the largest minimum distance of finite-length block codes. Numerical examinations show that the new lower bound improves the finite-length Gilbert-Varshamov lower bound and can reach the minimum distance of existing finite-length block codes.

Many good quantum error-correcting codes were constructed from cyclic codes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we construct nonbinary quantum cyclic codes and asymmetric quantum cyclic codes that are derived from repeated-root cyclic codes for an arbitrary length ps, and determine the true minimum distance of all those codes. Some proposed quantum cyclic codes are optimal. Additionally, some proposed asymmetric quantum cyclic codes have better parameters than the ones available in the literature.

A new upper bound on the minimum distance of binary cyclic arithmetic codes of composite length is derived. New classes of binary cyclic arithmetic codes of composite length are introduced. The error correction capability of these codes is discussed, and in some cases the actual minimum distance is found. Decoding algorithms based on majority-logic decision are proposed for these codes.

Let $q$ be a power of 2 and $\gamma\le q$ an integer. Based on the codewords of $[q,2,q-1]$ extended Reed-Solomon (RS) code over the finite field $\mathbb {F}_q$ , we can construct a $(\gamma,q)$ -regular low-density parity-check (LDPC) code, called a full-length RS-LDPC code and denoted by $\mathcal{C}(\gamma,q)$ . In this letter, the minimum distance of these codes is investigated. For any given $q$ and $\gamma\le q$ , an upper bound on $d(\mathcal{C}(\gamma,q))$ , the minimum distance of $\mathcal{C}(\gamma,q)$ , is provided. Furthermore, we determine the values of $d(\mathcal{C}(\gamma,q))$ for $\gamma=2$ , 3, and 4, and present the closed-form expressions on the numbers of minimum-weight codewords in $\mathcal{C}(\gamma,q)$ for $\gamma=2$ and 3.

We propose a technique for improving LP decoding, based on the merging of check nodes. This technique can be applied to standard as well as generalized LDPC codes. Furthermore, we show how a recently-discovered linear-complexity LP decoder can be used to derive non-trivial lower bounds on the minimum distance of specific LDPC codes, with complexity that exhibits quadratic growth with respect to the block length. This bound can be refined using the check node merging technique. The lower bound on the minimum distance is shown to be an upper bound on the fractional distance of the code.

Toric codes are a class of m-dimensional cyclic codes introduced recently by Hansen (Coding theory, cryptography and related areas (Guanajuato, 1998), pp 132–142, Springer, Berlin, 2000; Appl Algebra Eng Commun Comput 13:289–300, 2002), and studied in Joyner (Appl Algebra Eng Commun Comput 15:63–79, 2004) and Little and Schenck (SIAM Discrete Math, 2007). They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope $$P \subseteq {\mathbb{R}}^m$$. As such, they are in a sense a natural extension of Reed–Solomon codes. Several articles cited above use intersection theory on toric varieties to derive bounds on the minimum distance of some toric codes. In this paper, we will provide a more elementary approach that applies equally well to many toric codes for all $$m \ge 2$$. Our methods are based on a sort of multivariate generalization of Vandermonde determinants that has also been used in the study of multivariate polynomial interpolation. We use these Vandermonde determinants to determine the minimum distance of toric codes from simplices and rectangular polytopes. We also prove a general result showing that if there is a unimodular integer affine transformation taking one polytope P 1 to a second polytope P 2, then the corresponding toric codes are monomially equivalent (hence have the same parameters). We use this to begin a classification of two-dimensional cyclic toric codes with small dimension.

Finding the minimum distance of linear codes is in general a NP-hard problem, we propose an efficient algorithm to attack this problem. The principle of this approach is to search code words locally around the all-zero code word perturbed by a level of noise magnitude, in other words the maximum of noise that can be corrected by a Soft-In decoder, anticipating that the resultant nearest non-zero code words will most likely contain the minimum Hamming weight code word, whose Hamming weight is equal to the minimum distance of the linear code. A numerous results prove that the proposed algorithm is valid for general linear codes and it is very fast comparing to all others known techniques, therefore it is a good tool for computing. Comparing to Joanna's works, we proof that our algorithm has a low complexity with a fast time of execution. For some linear RQs, QDCs and BCHs codes with unknown minimum distance, we give a good estimation (true) of the minimum distance where the length is less than 439.

Constructing quantum low-density parity-check (LDPC) codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large $X$-distance but a much smaller $Z$-distance. However, together with a classical expander LDPC code and a tensoring method that generalizes a construction of Hastings and also the Tillich--Zémor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$. We then exploit the expansion properties of the complex to devise the first polynomial-time algorithm that decodes above the square root barrier for quantum LDPC codes. Using a 3-dimensional Ramanujan complex, we also obtain an overall quantum code of minimum distance $n^{1/2}\log n$, which sets a new record for quantum LDPC codes.

Twisted permutation codes, introduced recently by the second and third authors, belong to the family of frequency permutation arrays. Like some other codes in this family, such as the repetition permutation codes, they are obtained by a repetition construction applied to a smaller code (but with a "twist" allowed). The minimum distance of a twisted permutation code is known to be at least the minimum distance of a corresponding repetition permutation code, but in some instances can be larger. We construct two new infinite families of twisted permutation codes with minimum distances strictly greater than those for the corresponding repetition permutation codes. These constructions are based on two infinite families of finite groups and their representations. The first is a family of $p$-groups, for an odd prime $p$, while the second family consists of the $4$-dimensional symplectic groups over a finite field of even order. In the latter construction, properties of the graph automorphism of these symplectic groups play an important role.

Two <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -digit sequences, called "points," of binary digits are said to be at distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> if exactly <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> corresponding digits are unlike in the two sequences. The construction of sets of points, called codes, in which some specified minimum distance is maintained between pairs of points is of interest in the design of self-checking systems for computing with or transmitting binary digits, the minimum distance being the minimum number of digital errors required to produce an undetected error in the system output. Previous work in the field had established general upper bounds for the number of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -digit points in codes of minimum distance d with certain properties. This paper gives new results in the field in the form of theorems which permit systematic construction of codes for given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, d</tex> ; for some <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, d</tex> , the codes contain the greatest possible numbers of points.

Some new lower bounds on the minimum distance of cyclic codes of composite length are presented. These bounds generalize some existing bounds and give better estimates of the minimum distance for a wide range of cyclic codes, Lower bounds on the random-error-correction capability of a particular class of cyclic codes of composite length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n = n_1n_2</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_1</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_2</tex> are relatively prime, are easily derived.

An upper bound on the minimum distance of cyclic codes of composite length is presented. This upper bound proves the BCH bound to be exact for many cyclic codes.

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary k-dimensional linear code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ( k = 3,4) into a self-orthogonal code of the shortest length which has the same dimension k and minimum distance d' ≥ d( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ). For k > 4, we suggest a recursive method to embed a k-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length n with dimension 4 and any length n \not ≡ 6, 13,14,21,22,28,29 (mod 31) with dimension 5.

A characterization of MDS codes that have an error correcting pair