An additive quaternary [n,k,d]-code (length n, quaternary dimension k, minimum distance d) is a 2k-dimensional \mathbb F2-vector space of n-tuples with entries in \mathbb F2⊕\mathbb F2(the 2-dimensional vector space over \mathbb F2) with minimum Hamming distance d. We determine the optimal parameters of additive quaternary codes of dimension k ≤ 3. The most challenging case is dimension k=2.5. We prove that an additive quaternary [n,2.5,d]-code where d2-code for e <; m-1 exists if and only if the Griesmer bound 3(m-e) ≥ e/2+e/4+e/8 is satisfied.

We consider the [ q+1, q-3,5]q3 generalized doubly-extended Reed-Solomon code of codimension 4 as the code associated with the twisted cubic in the projective space PG(3,q). Basing on the point-plane incidence matrix of PG(3,q), we obtain the number of weight 3 vectors in all the cosets of the considered code. This allows us to classify the cosets by their weight distributions and to obtain these distributions. The weight of a coset is the smallest Hamming weight of any vector in the coset. For the cosets of equal weight having distinct weight distributions, we prove that the difference between the w-th components, 3 <; w ≤ q+1, of the distributions is uniquely determined by the difference between the 3-rd components. This implies an interesting (and in some sense unexpected) symmetry of the obtained distributions.

We study additive quaternary codes whose parameters are close to those of the extended cyclic [12, 6, 6]4-code or to the quaternary linear codes generated by the elliptic quadric in PG(3, 4) or its dual. In particular we characterize those codes in the category of additive codes and construct some additive codes whose parameters are better than those of any linear quaternary code. Our new code parameters are [22, 17.5, 4]4.

One of the oldest problems in the theory of quantum stabilizer codes is solved by proving the nonexistence of quantum [[13,5,4]]-codes.

In this paper, the best parameters of quaternary additive codes of small length are determined using the geometric description. Only one open question remains for length les 13. Among the results obtained in this work are the nonexistence of [12, 7, 5]-codes and [12, 4.5, 7]-codes as well as the existence of a [13, 7.5, 5]-code.

We define a class of codes admitting a large automorphism group. This family contains the binary extended Hamming code, the hexacode, the Golay codes, the Pless symmetry codes, as well as the [16,4,12]/sub 8/-codes constructed by Marcugini et al. A computer search resulted in the construction of codes with new parameters [28,7,18]/sub 8/,[32,8,20]/sub 8/, and [39,13,17]/sub 4/ belonging to this family.

A concept of locally optimal (LO) linear covering codes is introduced in accordance with the concept of minimal saturating sets in projective spaces over finite fields. An LO code is nonshortening in the sense that one cannot remove any column from a parity-check matrix without increasing the code covering radius. Several q/sup m/-concatenating constructions of LO covering codes are described. Taking a starting LO code as a "seed", such constructions produce an infinite family of LO codes with the same covering radius. The infinite families of LO codes are designed using minimal saturating sets as starting codes. New upper bounds on the length function are given. New extremal and classification problems for linear covering codes are formulated and investigated, in particular, the spectrum of possible lengths of LO codes including the greatest possible length. The complete computer classification of the minimal saturating sets in small geometries and of the corresponding LO codes is obtained.

Infinite families of linear codes with covering radius R=2, 3 and codimension tR+1 are constructed on the base of starting codes with codimension 3 and 4. Parity-check matrices of the starting codes are treated as saturating sets in projective geometry that are obtained by computer search using projective properties of objects. Upper bounds on the length function and on the smallest sizes of saturating sets are given.

A linear [n,k,d]/sub q/ code C is called near maximum-distance separable (NMDS) if d(C)=n-k and d(C/sup /spl perp//)=k. The maximum length of an NMDS [n,k,d]/sub q/ code is denoted by m'(k,q). In this correspondence, it has been verified by a computer-based proof that m'(5,8)=15, m'(4,9)=16,m'(5,9)=16, and 20/spl les/m'(4,11)/spl les/21. Moreover, the NMDS codes of length m'(4,8), m'(5,8), and m'(4,9) have been classified. As the dual code of an NMDS code is NMDS, the values of m'(k,8), k=10,11,12, and of m'(k,9),k=12,13,14 have been also deduced.