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 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.

We give necessary and sufficient conditions for the existence of stabilizer codes$[[n,k,3]]$of distance 3 for qubits:$n-k\geq \lceil \log _{2}(3n+1)\rceil +\epsilon _{n}$, where$\epsilon _{n}=1$if$n=8 {{ 4^{m}-1}\over { 3}}+\{\pm 1,2\}$or$n= {{ 4^{m+2}-1}\over { 3}}-\{1,2,3\}$for some integer$m\geq 1$and$\epsilon _{n}=0$otherwise. Or equivalently, a code$[[n,n-r,3]]$exists if and only if$n\leq (4^{r}-1)/3, (4^{r}-1)/3-n\notin \lbrace 1,2,3\rbrace $for even$r$and$n\leq 8(4^{r-3}-1)/3, 8(4^{r-3}-1)/3-n\ne 1$for odd$r$. Given an arbitrary length$n$, we present an explicit construction for an optimal quantum stabilizer code of distance 3 that saturates the above bound.

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.

We introduce a computer-based method for extending linear codes, which can be viewed as an inverse of the familiar construction Y/sub 1/. As a result codes with record-breaking parameters are constructed.

We determine the dimensions of subfield codes of Reed-Solomon codes and construct certain extensions and lengthenings of these codes. We start from the duals, using the language of orthogonal arrays. As a first result this allows us to obtain a fair number of improvements in the list of binary, ternary, and quaternary linear codes with largest known minimal distance.

We use lengthening and an enhanced version of the Gilbert-Varshamov lower bound for linear codes to construct a large number of record-breaking codes. Our main theorem may be seen as a closure operation on databases.

Orthogonal arrays (OAs) are basic combinatorial structures, which appear under various disguises in cryptology and the theory of algorithms. Among their applications are universal hashing, authentication codes, resilient and correlation-immune functions, derandomization of algorithms, and perfect local randomizers. In this paper, we give new explicit bounds on the size of orthogonal arrays using Delsarte’s linear programming method. Specifically, we prove that the minimum number of rows in a binary orthogonal array of length n and strength t is at least $2^n - (n2^{n - 1} /t + 1)$ and also at least $2^n - ( 2^{n - 2} (n + 1)/t\lceil \frac{{t + 1}}{2} \rceil )$. We also prove that these bounds are as powerful as the linear programming bound itself for many parametric situations. An $(n,m,t)$-resilient function is a function $f:\{ 0,1\} ^n \to \{ 0,1\} ^m $ such that every possible output m-tuple is equally likely to occur when the values of t arbitrary inputs are fixed by an opponent and the remaining $n - t$ input bits are chosen independently at random. A basic problem is to maximize t given m and n, i.e., to determine the largest value of t such that an $(n,m,t)$-resilient function exists. In this paper, we obtain upper and lower bounds for the optimal values of t where $1 \leq n \leq 25$ and $1 \leq m < n$. The upper bounds are derived from Delsarte’s linear programming bound, and the lower bounds come from constructions based on error-correcting codes. We also obtain new explicit upper bounds for the optimal values of t. It was proved by Chor et al. in [Proc. 26th IEEE Symp. on Foundations of Computer Science, 1985, pp. 396–407] that an $(n,2,t)$-resilient function exists if and only if $t < \lfloor {\frac{{2n}}{3}} \rfloor $. This result was generalized by Friedman [Proc. 33rd IEEE Symp. on Foundations of Computer Science, 1992, pp. 314–319], who proved a bound for general m. We also prove some new bounds, and complete the determination of the optimal resiliency of resilient functions with $m = 3$ and most of the cases for $m = 4$. Several other infinite classes of (optimal) resilient functions are also constructed using the theory of anticodes.