Codes in the Grassmannian have recently found an application in random network coding. All the codewords in such codes are subspaces of Fq with a given dimension. In this paper, we consider the problem of list decoding of a certain family of codes in the Grassmannian, called lifted Gabidulin codes. For this purpose we use the Plücker embedding of the Grassmannian. We describe a way of represent...

We present a bound on the size of linear codes. This bound is independent of other known bounds, e.g. the Griesmer bound.

We consider a class of linear codes associated to projective algebraic varieties defined by the vanishing of minors of a fixed size of a generic matrix. It is seen that the resulting code has only a small number of distinct weights. The case of varieties defined by the vanishing of 2× 2 minors is considered in some detail. Here we obtain the complete weight distribution. Moreover, several gener...

This paper provides an alternative proof to the MacWilliams identity for linear block codes over GF(q). The intermediate results of our alternative proof offer some insights of the vector-space structure. Our vector-space-based methodology can be used to establish the relation between weight distributions of dual codes for other metrics.

Let k3(n) denote the minimal cardinality of a ternary code of length n and covering radius one. In this paper we show k3(7) ≥ 156 and k3(8) ≥ 402 improving on the best previously known bounds k3(7) ≥ 153 and k3(8) ≥ 398. The proofs are founded on a recent technique of the author for dealing with systems of linear inequalities satisfied by the number of elements of a covering code, that lie in k...

We prove the nonexistence of linear codes with parameters [400; 5; 299]4, [401; 5; 300]4, [405; 5; 303]4, [406; 5; 304]4, [485; 5; 363]4 and [486; 5; 364]4 attaining the Griesmer bound. For that purpose we give a characterization of linear codes with parameters [86; 4; 64]4, [101; 4; 75]4, [102; 4; 76]4 and [122; 4; 91]4. c © 2001 Elsevier Science B.V. All rights reserved.

When there exists a malicious attacker in the network, we need to consider possibilities of eavesdropping and contamination simultaneously. Under an acyclic broadcast optimality linear codes was shown when Eve is allowed attack any $r$ edges. The not under different assumption for Eve. As typical example unicast focus on one-hop relay network single transmission scheme by assuming that attacks ...

In this paper, we employ the concept of the Generalized Discrete Fourier Transform, which in turn relies on the Hasse derivative of polynomials, to give a general construction of Reed-Solomon codes over Galois fields of characteristic not necessarily co-prime with the length of the code. The constructed linear codes  enjoy nice algebraic properties just as the classic one.