Bounds on the minimum distance of the duals of BCH codes

International audienceWe consider duals of BCH codes of length p^m-1 over GF(p). A lower bound on their minimum distance is found via the adaptation of the Weil bound to cyclic codes. However, this bound is of no significance for roughly half of these codes. We partially fill this gap by giving a lower bound for an infinite class of duals of BCH codes. We also present a lower bound obtained with an algorithm due to Massey and Schaub (1988). In the case of binary codes of length 127 and 255, the results are surprisingly higher than all previously known bound

On Quantum and Classical BCH Codes

Classical BCH codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length n can contain its dual code only if its designed distance d=O(sqrt(n)), and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and - consequently - the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters.Comment: 17 pages, LaTe

On spectra of BCH codes

Derives an estimate for the error term in the binomial approximation of spectra of BCH codes. This estimate asymptotically improves on the bounds by Sidelnikov (1971), Kasami et al. (1985), and Sole (1990)

On Primitive BCH Codes with Unequal Error Protection Capabilities

Presents a class of binary primitive BCH codes that have unequal-error-protection (UEP) capabilities. The authors use a previous result on the span of their minimum weight vectors to show that binary primitive BCH codes, containing second-order punctured Reed-Muller (RM) codes of the same minimum distance, are binary-cyclic UEP codes. The values of the error correction levels for this class of binary LUEP codes are estimated

Higher weight distribution of linearized Reed-Solomon codes

Linearized Reed-Solomon codes are defined. Higher weight distribution of those codes are determined

Linear codes over Fq which are equivalent to LCD codes

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general construction of LCD codes from any linear codes. Further, we show that any linear code over $\mathbb F_{q} (q>3)$ is equivalent to an Euclidean LCD code and any linear code over $\mathbb F_{q^2} (q>2)$ is equivalent to a Hermitian LCD code. Consequently an $[n,k,d]$-linear Euclidean LCD code over $\mathbb F_q$ with $q>3$ exists if there is an $[n,k,d]$-linear code over $\mathbb F_q$ and an $[n,k,d]$-linear Hermitian LCD code over $\mathbb F_{q^2}$ with $q>2$ exists if there is an $[n,k,d]$-linear code over $\mathbb F_{q^2}$. Hence, when $q>3$ (resp.$q>2$) $q$-ary Euclidean (resp. $q^2$-ary Hermitian) LCD codes possess the same asymptotical bound as $q$-ary linear codes (resp. $q^2$-ary linear codes). Finally, we present an approach of constructing LCD codes by extending linear codes.Comment: arXiv admin note: text overlap with arXiv:1702.0803

A note on the duals of skew constacyclic codes

Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $\theta : \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with skew constacyclic codes, that is, linear codes of $\mathbb{F}_q^n$ which are invariant under the action of a semi-linear map $\Phi_{\alpha,\theta}:\mathbb{F}_q^n\to\mathbb{F}_q^n$, defined by $\Phi_{\alpha,\theta}(a_0,...,a_{n-2}, a_{n-1}):=(\alpha \theta(a_{n-1}),\theta(a_0),...,\theta(a_{n-2}))$ for some $\alpha\in\mathbb{F}_q\setminus\{0\}$ and $n\geq 2$. In particular, we study some algebraic and geometric properties of their dual codes and we give some consequences and research results on $1$-generator skew quasi-twisted codes and on MDS skew constacyclic codes.Comment: 31 pages, 3 tables; this is a revised version that includes improvements to the presentation of the main results, a new subsection and an appendix which is an extension of Section 2 of the previous versio

Enlargement of Calderbank Shor Steane quantum codes

It is shown that a classical error correcting code C = [n,k,d] which contains its dual, C^{\perp} \subseteq C, and which can be enlarged to C' = [n,k' > k+1, d'], can be converted into a quantum code of parameters [[ n, k+k' - n, min(d, 3d'/2) ]]. This is a generalisation of a previous construction, it enables many new codes of good efficiency to be discovered. Examples based on classical Bose Chaudhuri Hocquenghem (BCH) codes are discussed.Comment: 12 pages. Submitted to IEEE Trans. Inf. Theory. Mistake in proof correcte

Decoding of Repeated-Root Cyclic Codes up to New Bounds on Their Minimum Distance

The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic codes and present a syndrome-based burst error decoding algorithm with guaranteed decoding radius based on an associated folded cyclic code. Furthermore, we present a third technique for bounding the minimum Hamming distance based on the embedding of a given repeated-root cyclic code into a repeated-root cyclic product code. A second quadratic-time probabilistic burst error decoding procedure based on the third bound is outlined. Index Terms Bound on the minimum distance, burst error, efficient decoding, folded code, repeated-root cyclic code, repeated-root cyclic product cod

Weight distribution of cosets of small codes with good dual properties

The bilateral minimum distance of a binary linear code is the maximum $d$ such that all nonzero codewords have weights between $d$ and $n-d$. Let $Q\subset \{0,1\}^n$ be a binary linear code whose dual has bilateral minimum distance at least $d$, where $d$ is odd. Roughly speaking, we show that the average $L_\infty$-distance -- and consequently the $L_1$-distance -- between the weight distribution of a random cosets of $Q$ and the binomial distribution decays quickly as the bilateral minimum distance $d$ of the dual of $Q$ increases. For $d = \Theta(1)$, it decays like $n^{-\Theta(d)}$. On the other $d=\Theta(n)$ extreme, it decays like and $e^{-\Theta(d)}$. It follows that, almost all cosets of $Q$ have weight distributions very close to the to the binomial distribution. In particular, we establish the following bounds. If the dual of $Q$ has bilateral minimum distance at least $d=2t+1$, where $t\geq 1$ is an integer, then the average $L_\infty$-distance is at most $\min\{\left(e\ln{\frac{n}{2t}}\right)^{t}\left(\frac{2t}{n}\right)^{\frac{t}{2} }, \sqrt{2} e^{-\frac{t}{10}}\}$. For the average $L_1$-distance, we conclude the bound $\min\{(2t+1)\left(e\ln{\frac{n}{2t}}\right)^{t} \left(\frac{2t}{n}\right)^{\frac{t}{2}-1},\sqrt{2}(n+1)e^{-\frac{t}{10}}\}$, which gives nontrivial results for $t\geq 3$. We given applications to the weight distribution of cosets of extended Hadamard codes and extended dual BCH codes. Our argument is based on Fourier analysis, linear programming, and polynomial approximation techniques