Identities and approximations for the weight distribution of q-ary codes

An explicit formula is derived that enumerates the complete weight distribution of an (n, k, d) linear code using a partially known weight distribution. An approximation formula for the weight distribution of q-ary linear (n, k , d) codes is also derived. It is shown that, for a given q-ary linear (n, k, d) code, the ratio of the number of codewords of weight u to the number of words of weight u approaches the constant Q=q ^-(n-k) as u becomes large. The error term is a decreasing function of the minimum weight of the dual. The results are also valid for nonlinear (n, M, d) codes with the minimum weight of the dual replaced by the dual distance     

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

n_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 n₄(4,d) is determined for all but 52 values of d     

Pseudocyclic maximum-distance-separable codes

The (n, k) pseudocyclic maximum-distance-separable (MDS) codes modulo (xⁿ- a) over GF(q) are considered. Suppose that n is a divisor of q+1. If n is odd, pseudocyclic MDS codes exist for all k. However, if n is even, nontrivial pseudocyclic MDS codes exist for odd k (but not for even k) if a is a quadratic residue in GF(q), and they exist for even k (but not for odd k) if a is not a quadratic residue in GF(q). Also considered is the case when n is a divisor of q-1, and it is shown that pseudocyclic MDS codes exist if and only if the multiplicative order of a divides (q-1)/n, and that when this condition is satisfied, such codes exist for all k. If the condition is not satisfied, every pseudocyclic code of length n is the result of interleaving a shorter pseudocyclic code     

Composition of Reed-Solomon codes and geometric designs

It is shown that good linear (n,k,d) codes over a finite field GF(q) can be constructed by concatenating the generator matrices of Reed-Solomon codes. For the case of k=3, it is shown that many of the codes obtained using projective-geometry techniques can readily be obtained by the proposed algebraic approach     

On repeated-root cyclic codes

A parity-check matrix for a q-ary repeated-root cyclic code is derived using the Hasse derivative. Then the minimum distance of a q-ary repeated-root cyclic code is expressed in terms of the minimum distance of a certain simple-root cyclic code. With the help of this result, several binary repeated-root cyclic codes of lengths up to n=62 are shown to contain the largest known number of codewords for their given length and minimum distance. The relative minimum distance d_min/n of q-ary repeated-root cyclic codes of rate r⩾R is proven to tend to zero as the largest multiplicity of a root of the generator g(x) increases to infinity. It is further shown that repeated-root cycle codes cannot be asymptotically better than simple-root cyclic codes     

New minimum distance bounds for certain binary linear codes

Let an [n, k, d]-code denote a binary linear code of length n, dimension k, and minimum distance at least d. Define d(n, k) as the maximum value of d for which there exists a binary linear [n, k, d]-code. T. Verhoeff (1989) has provided an updated table of bounds on d(n, k) for 1⩽k⩽n⩽127. The authors improve on some of the upper bounds given in that table by proving the nonexistence of codes with certain parameters     

Some new upper bounds on the covering radius of binary linear codes

A Griesmer-like upper bound on the covering radius, R, is given. To the author's knowledge this is the only upper bound which explicitly depends on all three parameters n, k, and d. An upper bound on R for cyclic codes is then given which depends on the generator polynomial of the cyclic code and which, in many cases, leads to an improvement of the previous bound. An upper bound on the irreducible generator polynomial cyclic codes is also given. New interpretations and applications of the so-called Norse bounds and necessary and sufficient conditions to attain one of these bounds are provided. Generalizations of most of the results for codes over GF(q) are outlined     

A general decoding technique applicable to replicated filedisagreement location and concatenated code decoding

Code symbols are treated as vectors in an r-dimensional vector space F^r over a field F. Given any ( n, k) linear block code over F with minimum distance d, it is possible to derive an (n, k) code with symbols over F^r, also with minimum distance d, which can correct any pattern of d-2 or fewer symbol errors for which the symbol errors as vectors are linearly independent. This is about twice the bound on the number of errors guaranteed to be correctable. Furthermore, if the error vectors are linearly dependent and d-2 or fewer in number, the existence of dependence can always be detected. A decoding techinque is described for which complexity increases no greater than as n ³, for any choice of code. For the two applications considered, situations are described where the probability of the error patterns being linearly dependent decreases exponentially with r     

On the decoding of algebraic-geometric codes

A decoding algorithm for algebraic-geometric codes arising from arbitrary algebraic curves is presented. This algorithm corrects any number of errors up to [(d-g-1)/2], where d is the designed distance of the code and g is the genus of the curve. The complexity of decoding equals σ(n³) where n is the length of the code. Also presented is a modification of this algorithm, which in the case of elliptic and hyperelliptic curves is able to correct [(d-1)/2] errors. It is shown that for some codes based on plane curves the modified decoding algorithm corrects approximately d/2-g/4 errors. Asymptotically good q-ary codes with a polynomial construction and a polynomial decoding algorithm (for q⩾361 on some segment their parameters are better than the Gilbert-Varshamov bound) are obtained. A family of asymptotically good binary codes with polynomial construction and polynomial decoding is also obtained, whose parameters are better than the Blokh-Zyablov bound on the whole interval 0<σ<1/2     

Binary [18,11]2 codes do not exist-Nor do [64,53]2 codes

A binary, linear block code C with block length n and dimension n is commonly denoted by [n, k] or, if its minimum distance is d, by [n, k,d]. The code's covering radius r(C) can be defined as the smallest number r such that any binary column vector of length (n-k) can be written as a sum of r or fewer columns of a parity-check matrix of C. An [n,k] code with covering radius r is denoted by [n,k]r. R.A. Brualdi et al., (1989) showed that l(m,r) is defined to be the smallest n such that an [n,n-m]r code exists. l(m,2) is known for m⩽6, while it is shown by Brualdi et al. that 17⩽l(7,2)⩽19. This lower bound is improved by A.R. Calderbank et al. (1988), where it is shown that [17,10]2 codes do not exist. The nonexistence of [18,11]2 codes is proved, so that l(7,2)=19. l[7.2)=19 is established by showing that [18,11]2 codes do not exist. It is also shown that [64,53]2 codes do not exist, implying that l(11,2)⩾65     

A construction of non-Reed-Solomon type MDS codes

A construction is presented of long maximum-distance-separable (MDS) codes that are not generalized Reed-Solomon (GRS) type. The construction uses subsets S,|S|=m of a finite field F=GF(q) with the property that no t distinct elements of S add up to some fixed element of F . Large subsets of this kind are used to construct [n=m+2, k=t+1] non-GRS MDS codes over F     

On the existence of optimum cyclic burst correcting codes over GF(q)

A cyclic b-burst correcting code over GF(q) of redundancy r and length n=(q^r-b+1-1)/(q-1) is said to be optimum. It is proved that a necessary condition for the existence of such a code is the existence of a square-free polynomial in GF(q)[x] of degree b-1 which is not divisible by x such that its period and the degrees of its irreducible factors are relatively prime to q-1. Moreover, if such a polynomial exists, then there are an infinite number of optimum cyclic b-burst correcting codes over GF(q)     

Lower bounds for q-ary covering codes

The authors prove combinatorial lower bounds for K_q (n,R), the minimal cardinality of any q-ary code of length n and covering radius R. Tables of lower bounds for K_q(n,R) are presented for q=3, 4, 5     

Relationships between m-sequences over GF(q) andGF(qm)

It is shown that m-sequences over GF(q^m ) of length q^nm-1 corresponding to primitive polynomials in GF[q^m,x] of degree n can be generated from known m-sequences over GF(q) of length q^nm-1 obtained from primitive polynomials in GF[q,x] of degree mn. A procedure for generating the m-sequences over GF(q²) from m-sequences over GF(q) was given which enables the generation of m-sequences over GF( p²ⁿ). In addition it was shown that all of the primitive polynomials in GF[q,^m,x] can be obtained from a complete set of the primitive polynomials in GF[q ,x]     

Optimal codes for minimax criterion on error detection

Nonlinear quadratic codes that are optimal for the minimax error detection are presented. Characteristic functions for these codes are asymptotically bent. For a given block size n and the number of codewords |C|, these codes minimize max Q(e), e≠0, where Q(e) is the conditional error-masking probability, given the error pattern e. The codewords are blocks of n symbols from GF(q). Encoding and decoding procedures for the codes are described     

A coding scheme for m-out-of-n codes

A scheme for the construction of m-out-of-n codes based on the arithmetic coding technique is described. For appropriate values of n, k, and m, the scheme can be used to construct an (n,k) block code in which all the codewords are of weight m. Such codes are useful, for example, in providing perfect error detection capability in asymmetric channels such as optical communication links and laser disks. The encoding and decoding algorithms of the scheme perform simple arithmetic operations recursively, thereby facilitating the construction of codes with relatively long block sizes. The scheme also allows the construction of optimal or nearly optimal m-out-of-n codes for a wide range of block sizes limited only by the arithmetic precision used     

Binary linear quasi-perfect codes are normal

Whether quasi-perfect codes are normal is addressed. Let C be a code of length n, dimension k, covering radius R, and minimal distance d. It is proved that C is normal if d⩾2R-1. Hence all quasi-perfect codes are normal. Consequently, any [n,k ]R binary linear code with minimal distance d⩾2R-1 is normal     

On (k, t)-subnormal covering codes

The concept of a (k, t)-subnormal covering code is defined. It is discussed how an amalgamated-direct-sumlike construction can be used to combine such codes. The existence of optimal (q, n, M) 1 codes C is discussed such that by puncturing the first coordinate of C one obtains a code with (q, 1)-subnorm 2     

The (d,k) subcode of a linear block code

A simple technique employing linear block codes to construct (d,k) error-correcting block codes is considered. This scheme allows asymptotically reliable transmission at rate R over a BSC channel with capacity C_BSC provided R ⩽C_d,k-(1+C_BSC), where C_d,k is the maximum entropy of a (d,k ) source. For the same error-correcting capability, the loss in code rate incurred by a multiple-error correcting (d,k) code resulting from this scheme is no greater than that incurred by the parent linear block code. The single-error correcting code is asymptotically optimal. A modification allows the correction of single bit-shaft errors as well. Decoding can be accomplished using off-the-shelf decoders. A systematic (but suboptimal) encoding scheme and detailed case studies are provided