SOME MDS AND ACD CODES OVER COMMUTATIVE NON-UNITAL RINGS OF ORDERS 4 AND 9

초록

There are eleven finite rings of order p2 denoted by Ap to Kp in alphabetical order. In particular, we consider I2 and I3, which are commutative non-unital rings of orders 4 and 9 defined by generators and relations as for p = 2, 3, respectively. Alahmadi et al. studied codes over these rings. In this paper, we study additive complementary dual (ACD) codes over the rings I2 and I3. We show relations between ACD codes over I2 and binary linear complementary dual (LCD) codes using a reduction map from I2 to F2, and between ACD codes over I3 and ternary LCD codes using a reduction map from I3 to F3. Using the first relation, we classify ACD codes over I2 with the highest minimum distances for n = 1, 2,3 and partially for n = 4, 5. It turns out that they are maximum distance separable (MDS) codes. Using the second relation, we classify ACD codes over I3 with the highest minimum Lee distances for n = 1, 2 and partially for n = 3. We generalize the two relations into a relation between ACD codes over Ip and p-ary LCD codes using a reduction map from Ip to Fp.

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