Projective Reed-Solomon (PRS) codes are Reed-Solomon codes of the maximum possible length q+1. The classification of deep holes-received words with maximum possible error distance- for PRS codes is an important and difficult problem. In this paper, we use algebraic methods to explicitly construct three classes of deep holes for PRS codes. We show that these three classes completely classify all deep holes of PRS codes with redundancy four. Previously, the deep hole classification was only known for PRS codes with redundancy at most three.

For non-binary codes the Elias bound is a good upper bound for the asymptotic information rate at low-relative minimum distance, whereas the Plotkin bound is better at high-relative minimum distance. In this paper, we obtain a hybrid of these bounds, which improves both. This in turn is based on the anticode bound, which is a hybrid of the Hamming and Singleton bounds and improves both bounds. The question of convexity of the asymptotic rate function is an important open question. We conjecture a much weaker form of the convexity, and we show that our bounds follow immediately if we assume the conjecture.

We study the problem of classifying deep holes of Reed-Solomon codes. We show that this problem is equivalent to the problem of classifying maximum distance separable (MDS) extensions of Reed-Solomon codes by one digit. This equivalence allows us to improve recent results on the former problem. In particular, we classify deep holes of Reed-Solomon codes of dimension greater than half the alphabet size. We also give a complete classification of deep holes of Reed-Solomon codes with redundancy three in all dimensions.

We obtain an asymptotic formula in q, as well as new upper and lower bounds, for the number of MDS codes of length n and dimension k over a finite field with q elements.

In this paper, we consider the problem of determining the weight spectrum ofq-ary codesC(3,m) associated with Grassmann varietiesG(3,m). Form=6, this was done in a paper by Nogin in 1997. We derive a formula for the weight of a codeword ofC(3,m), in terms of certain varieties associated with alternating trilinear forms on Fqm. The classification of such forms under the action of the general linear groupGL(m, Fq) is the other component that is required to calculate the spectrum ofC(3,m). Form=7, we explicitly determine the varieties mentioned above. The classification problem for alternating three-forms on Fq7was solved in a study by Cohen and Helminck in 1988, which we then use to determine the spectrum ofC(3,7).