Q-ary Codes Research Articles

On a sequence of cyclic codes with minimum distance six

A sequence of q-ary cyclic codes is considered. For each finite field GF(q), q/spl ges/4, there is a code with parameters [n, k, d; q]=[q(q-1)+1, q(q-1)-6, 6; q]. We show that all these codes are n-, k-, and d-optimal, with only one exception. Also the dual codes are considered. Their true minimum distances are calculated in the range 4/spl les/q/spl les/32.

Linear Codes and Their Coordinate Ordering

A necessary and sufficient condition for a q-ary code to satisfy the two-way chain condition (TCC) is found. A known construction of q-ary codes is shown to yield codes satisfying the TCC. Some q-ary codes of dimension k \leq 6 meeting the Griesmer bound are proved to satisfy the TCC.

Two classes of q-ary codes based on group divisible association schemes

By means of Hall's Marriage Theorem, a class of q-ary codes are obtained through block designs based on group divisible association schemes with two associate classes. If these block designs are further semiframes, another class of q-ary codes are also constructed.

An Extension Theorem for Linear Codes

One of the first results one meets in coding theory is that a binary linear [n,k,d ] code, whose minimum distance is odd, can be extended to an [ n+1,k,d+1] code. This is one of the few elementary results about binary codes which does not obviously generalise to q-ary codes. The aim of this paper is to give a simple sufficient condition for a q-ary [ n,k,d] code to be extendable to an [ n+1,k,d+1] code. Applications will be given to the construction and classification of good codes, to proving the non- existence of certain codes, and also an application in finite geometry.

Decoder error and failure probabilities for Reed-Solomon codes: decodable vectors method

In this letter, formulas for the total number of decodable and undecodable vectors are derived for a general (n,k) q-ary linear block code. These formulas are used to find the probabilities of decoder error and decoder failure for Reed-Solomon codes under errors-and-erasures decoding. The resulting analytical expressions are computationally efficient and allow accurate calculation of very small values of decoder failure probabilities. The formulas are used to analyze the performance of type-I hybrid automatic-repeat-request (HARQ) protocols with two decoding diameters.

Some new bounds for q-ary linear block codes

In this correspondence, we derive, with the method of coset decompositions, some new bounds for q-ary linear block codes, where q is the order of some finite field.

Algoritmo Geral para a Procura De Códigos Treliça Ótimos Baseados em Partições de Reticulados

In this paper a general algorithm for the search of optimum q-ary trellis codes, based on lattices and cosets, is proposed. The algorithm incorporates the procedures introduced for binary codes in [2],[4] and generalizes the search for alphabets that are rings with q elements, Z_q. New characteristics were incorporated to the algorithm, providing a faster and more efficient search for the optimum codes than in [3],[4],[5]. Numerical examples are presented in details for binary scheme with the Z^2/\thetaZ^2 partition and for a ternary scheme with the A_2/9A_2 partition.

A Construction for (t,m,s)-nets in Base q

It has been shown that (t,m,s)-nets in base b can be characterized by combinatorial objects known as generalized orthogonal arrays. In this paper, a new construction for generalized orthogonal arrays leads to new (t,m,s)-nets in base q, q a prime power. The basic building block for the construction is an array of elements over Fq in which certain collections of rows are linearly independent. It is shown that if there exists an [n,n-m,d] q-ary code with $d\ge 6+2p$, where $p\ge 0$ is an integer, then there exists a (t,m,s)-net in base q with $t=m-(4+2p)$ and $s= \lfloor \frac{n-1}{1+p} \rfloor$.

On the Dimension of the Hull

The hull [Assmus, Jr. and Key, Discrete Math., 83 (1990), pp. 161--187], [Assmus, Jr. and Key, Designs and Their Codes, Cambridge University Press, 1992, p. 43] of a linear code is defined to be its intersection with its dual. We give here the number of distinct q-ary linear codes which have a hull of given dimension. We will prove that, asymptotically, the proportion of q-ary codes whose hull has dimension l is a positive constant that depends only on l and q and consequently that the average dimension of the hull is asymptotically a positive constant depending only on q.

A note on bounds for q-ary covering codes

Two strongly seminormal codes over Z/sub 5/ are constructed to prove a conjecture of Ostergard (see ibid., vol.37, no.3, p.660-4, 1991). It is shown that a result of Honkala (see ibid., vol.37, no.4, p.1203-6, 1991) on (k,t)-subnormal codes holds also under weaker assumptions. A lower bound and an upper bound on K/sub q/(n, R), the minimal cardinality of a q-ary code of length n with covering radius R are obtained. These give improvements in seven upper bounds and twelve lower bounds by Ostergard for K/sub q/(n, R) for q=3, 4, and 5.

McEliece public key cryptosystems using algebraic-geometric codes

McEliece (1978) proposed a public-key cryptosystem based on binary linear codes, in particular binary classical Goppa codes. We look at various aspects of McEliece's scheme in the general setting of q-ary codes. In particular, we consider schemes based on much larger class of q-ary algebraic-geometric (AG) Goppa codes, subfield subcodes of AG codes, and concatenated codes. We give explicit constructions of several schemes which have very high work factor, excellent key-length/plain-text ratios, and relatively smaller size of the keys for given work factors. We also present its modifications and generalizations following Krouk (see Proceedings of The Sixth Swedish-Russian International Workshop on Information Theory, p.285, 1993) and others. Finally, we discuss some open problems.

Suboptimal decoding of linear codes: partition technique

General symmetric channels are introduced, and near-maximum-likelihood decoding in these channels is studied. First, we define a class of suboptimal decoding algorithms based on an incomplete search through the code trellis. It is proved that the decoding error probability of suboptimal decoding is bounded above for any q-ary code of length n and code rate r by twice the error probability of its maximum-likelihood decoding and tends to the latter as n grows. Second, we design a suboptimal trellis-like algorithm, which reduces the known decoding complexity of the order of q/sup n min (r,1-r)/ operations to that of q/sup nr(i-r)/ operations for all cyclic codes and virtually all long linear codes. We also consider the corresponding bounds for concatenated codes. An important corollary is that this suboptimal decoding can provide complexity below the lower bounds on trellis complexity at a negligible expense in terms of decoding error probability.

Nonbinary double-error-correcting codes designed by means of algebraic varieties

Linear q-ary codes of growing length n/spl rarr//spl infin/ and designed distance /spl delta/ are studied. At first, we examine cyclic codes defined by the sets of code zeros {g/sup i/|i=q/sup s/+1, q/sup s+1/+1, /spl middot//spl middot//spl middot/, q/sup s+/spl delta/-2/+1} over a primitive element g of GF(q/sup m/). Then special cubic varieties are designed and employed in order to attain distances /spl delta/=5, 6. The resulting double-error-correcting codes of length n=q/sup m/ have r/spl les/2m+[m/3]+1 parity check symbols, and reduce the best known redundancy by [2m/3] symbols. A decoding procedure of complexity O(rn) operations is also considered.

On ( q, 1)-subnormal q-ary covering codes

We show that if q ≠ 3 is a prime power and there exists a ( q, n, M) 1 code, i.e., a q-ary code of length n with M codewords and covering radius 1 then there exists also a ( q, 1)-subnormal ( q, qn + 1, q ( q-1) n M)1 code. We also show that all nontrivial linear q-ary codes with covering radius 1 are ( q, 1)-subnormal with the exception of the ternary [4, 2]1 Hamming code.

A class of q-ary codes

A class of q-ary ( q⩾3) codes is constructed from nested balanced incomplete block designs.

Some new lower bounds for binary and ternary covering codes

A modified lower bound for binary codes with covering radius one is derived. Let K/sub q/(n,R) denote the minimum cardinality of a q-ary code with covering radius R. The new bound shows that K/sub 2/(11,1)>or=177 and that K/sub 2/(17,1)>or=7391, improvements of the best lower bounds known. The authors also generalize a known lower bound for binary codes to the case of arbitrary q. For q=3, this simple bound improves the best lower bounds known in several cases. An updated version of a table for K/sub 3/(n,R) is included.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The automorphism group of Generalized Reed-Muller codes

We prove that the automorphism group of Generalized Reed-Muller codes is the general linear nonhomogeneous group. The Generalized Reed-Muller codes are introduced by Kasami, Lin and Peterson. An extensive study was made by Delsarte, Goethals and Mac-Williams; our result follows their description of the minimum weight codewords. An automorphism of a cyclic q-ary code is here a substitution over the field GF( q m ). In the more general case where the automorphisms are defined by monomial matrices, we also obtain the automorphism group (called the monomial group) as the direct product of the general linear nonhomogeneous group with the multiplicative group of the alphabet field.

On Generator Matrices of Codes

The if class of the q-ary linear codes of given length, dimension and minimum weight is nonempty, it is shown to contain a code whose generator matrix consists of words of minimum weight.

Optimum codes of dimension 3 and 4 over GF(4)

n/sub q/(k,d), the length of a q-ary optimum code for given k and d, for q=4 and k=3, 4 is discussed. The problem is completely solved for k=3, and the exact value of n/sub 4/(4,d) is determined for all but 52 values of d. >

Bounds on packings and coverings by spheres in q-ary and mixed Hamming spaces

In a recent paper by the same author general improvements on the sphere covering bound for binary covering codes were obtained. In the present work it is shown how the main idea can be exploited to obtain “improved sphere bounds” also for nonbinary codes. We concentrate on general q-ary codes and on binary/ternary mixed codes. Special attention is paid to the football pool problem; a few new lower bounds are established. Also, it is shown how a similar method yields upper bounds on packing codes.