Self-dual codes and orthogonal designs have been studied for a long time as separate research areas. In the present paper we show a strong relationship between them. The structure of orthogonal designs is such as to allow us a much faster and more systematic search for self-dual codes over GF(5). Using our method we constructed the following linear self-dual codes over GF(5);(i) [4,2,2], (ii) [8,4,4], (iii) [12,6,6], (iv) [16,8,6], (v) [20,10,8], (vi) [24,12,9], (vii) [28,14,10]. The codes (i), (ii), (iii), (v) are extremal. A [28,14,10] code is constructed here for the first time.
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