Lcd codes over F2+uF2 with small dimensions

초록

There are four commutative unital rings of order four. Among them, we consider the ring R = F2 + uF2 = {0, 1, u, >u = u + 1} where u2 = 0, which is a commutative ring with characteristic 2. In this paper, we study linear complementary dual (LCD) codes over the ring R. We first define LCD[n, k] R, which denotes the maximum of possible values of d among free [n, k, d] LCD codes over R, and obtain a Griesmer type bound for linear codes over R. We get an upper bound for LCD[n, 2] R, and further show that LCD[n, 2] R with the exception of n = 0,-1 (mod 6) meets the upper bound exactly. For k = 3, we also get an upper bound for LCD[n, 3] R. Then we show that LCD[n, 3] R meets the upper bound exactly for n = 3,5 (mod 7). We also derive bounds of LCD[n, k] R for k = 4, 5 from the binary cases. Furthermore, we obtain the exact value of LCD[n, n - i] R for i greater than or equal to two using the sphere packing bound.

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