A linear code with a complementary dual (or LCD code) is defined to be a linear code C whose dual code

We derive twenty five basic identities of symmetry in three variables related to higherorder Euler polynomials and alternating power sums. This demonstrates that there are abundant identities of symmetry in three-variable case, in contrast to two-variable case, where there are only a few. These are all new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the higher-order Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating power sums.

A secret sharing scheme (SSS) was introduced by Shamir in 1979 using polynomial interpolation. Later it turned out that it is equivalent to an SSS based on a Reed-Solomon code. SSSs based on linear codes have been studied by many researchers. However there is little research on SSSs based on additive codes. In this paper, we study SSSs based on additive codes over GF (4) and show that they require at least two steps of calculations to reveal the secret. We also define minimal access structures of SSSs from additive codes over GF (4) and describe SSSs using some interesting additive codes over GF (4) which contain generalized 2-designs.

As far as we know, there is no decoding algorithm of any binary self-dual [40,20,8] code except for the syndrome decoding applied to the code directly. This syndrome decoding for a binary self-dual [40,20,8] code is not efficient in the sense that it cannot be done by hand due to a large syndrome table. The purpose of this paper is to give two new efficient decoding algorithms for an extremal binary doubly-even self-dual [40,20,8] code C DE 40,1 by hand with the help of a Hermitian self-dual [10, 5, 4] code E 10 over GF (4). The main idea of this decoding is to project codewords of C DE 40,1 onto E 10 so that it reduces the complexity of the decoding of C DE 40,1 . The first algorithm is called the representation decoding algorithm. It is based on the pattern of codewords of E 10 . Using certain automorphisms of E 10 , we show that only eight types of codewords of E 10 can produce all the codewords of E 10 . The second algorithm is called the syndrome decoding algorithm based on E 10 . It first solves the syndrome equation in E 10 and finds a corresponding binary codeword of C DE 40,1 .

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