On optimal quantum codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters [[n,n-2d+2,d]]_q exist for all 3 <= n <= q and 1 <= d <= n/2+1. We also present quantum MDS codes with parameters [[q^2,q^2-2d+2,d]]_q for 1 <= d <= q which additionally give rise to shortened codes [[q^2-s,q^2-2d+2-s,d]]_q for some s.Comment: Accepted for publication in the International Journal of Quantum Informatio

Maximum Distance Separable Codes and Arcs in Projective Spaces

Given any linear code $C$ over a finite field $GF(q)$ we show how $C$ can be described in a transparent and geometrical way by using the associated Bruen-Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes.Comment: 18 Pages; co-author added; some results updated; references adde

The Weights in MDS Codes

The weights in MDS codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n-k+1 to n. The proof uses the covering radius of the dual codeComment: 5 pages, submitted to IEEE Trans. IT. This version 2 is the revised version after the refereeing process. Accepted for publicatio

On the decoder error probability for Reed-Solomon codes

Upper bounds On the decoder error probability for Reed-Solomon codes are derived. By definition, "decoder error" occurs when the decoder finds a codeword other than the transitted codeword; this is in contrast to "decoder failure," which occurs when the decoder fails to find any codeword at all. These results imply, for example, that for a t error-correcting Reed-Solomon code of length q - 1 over GF(q), if more than t errors occur, the probability of decoder error is less than 1/t!