dimensional subspace in V (n; q) such that every nonzero vector in C has a. Hamming weight ..... 1 0 a1 a2 b1 b2 c1 c2 d1 d2 e1 e2 f1 f2 g1 g2 h1 h2 i1 i2. 0 1 1. 2 1 2 1 2 1 ..... 2 = (13 14 15 16 18 17 1 2 4 3 5 6 7 8 10 9 12 11 ): (For instance, 2 ...
There are exactly two nonequivalent [20,5,12;3]-codes Noboru Hamada, Tor Helleseth, and Øyvind Ytrehus 1. Introduction Abstract — Hill and Newton showed that there exists a [20; 5; 12; 3]-code, and that the weight distribution of a [20; 5; 12; 3]-code is unique. However, it is unknown whether or not a code with these parameters is unique. Recently, Hamada and Helleseth showed that a [19; 4; 12; 3]-code is unique up to equivalence, and characterized this code using a characterization of f21; 6; 3; 3gminihypers. The purpose of this paper is to show, using the geometrical structure of the [19; 4; 12; 3]-code, that exactly two non-isomorphic [20; 5; 12; 3]-codes exist. Let V (n; q ) be an n-dimensional vector space over GF (q). If C is a kdimensional subspace in V (n; q) such that every nonzero vector in C has a Hamming weight (i. e., number of nonzero coordinates) of at least d, then C is denoted an [n; k; d; q] -code. The well-known Griesmer bound [Griesmer, 1960, Solomon and Stiffler, 1965] states that
n
k0 1 X d i=0
qi
(1.1)
where dxe denotes the smallest integer x. A coding theory problem that has been the subject of considerable research is the following: Main Problem. Characterize all [n; k; d; q] -codes meeting bound (1.1) with equality. Hill and Newton recently described a [20; 5; 12; 3]-code. In this paper we show that, up to equivalence, there are two types of [20; 5; 12; 3]-codes.
2. Preliminary results It is easy to show that if an [n; k; d qk01 ; q]-code meets bound (1.1) with equality, then any two column vectors of a generator matrix of the code must be linearly independent over GF (q). Thus, in this case, it can be convenient 1
to think of the set of columns of a generator matrix as a set of points in the finite projective geometry P G(k 0 1; q). Each column vector (c0 ; . . . ; ck01 )T represents a point P ,
0
T
(c0 ; . . . ; ck 1 )
$P =
01 X
!
k
ci i
(2.1)
i=0
where (0 ); . . . ; (k01 ) are k linearly independent, arbitrarily chosen points in P G(k 0 1; q). Since the point (a ) is equal to ( ) for any nonzero element a 2 GF (q), we need to consider as potential generator matrix columns only those vectors (c0 ; . . . ; ck 01 )T that have “1” as its last nonzero entry, i. e. that satisfy
9i : 0 i k 0 1 :
cj = 0; ci = 1
i
