Weight distributions of linear codes and the Gleason-Pierce theorem
Journal of Combinatorial Theory Series A
On designs and formally self-dual codes
Designs, Codes and Cryptography
The divisible code bound revisited
Journal of Combinatorial Theory Series A
On type IV self-dual codes over Z4
Discrete Mathematics
Higher Weights for Ternary and Quaternary Self-Dual Codes*
Designs, Codes and Cryptography
Quantum error correction via codes over GF(4)
IEEE Transactions on Information Theory
Type IV self-dual codes over rings
IEEE Transactions on Information Theory
Construction of optimal Type IV self-dual codes over F2+uF2
IEEE Transactions on Information Theory
Some results on type IV codes over Z4
IEEE Transactions on Information Theory
Optimal self-dual codes over F2×F2 with respect to the Hamming weight
IEEE Transactions on Information Theory
A note on formally self-dual even codes of length divisible by 8
Finite Fields and Their Applications
Classification of Type IV Self-Dual Z4-Codes of Length 16
Finite Fields and Their Applications
On the classification and enumeration of self-dual codes
Finite Fields and Their Applications
An equivalence of Ward's bound and its application
Designs, Codes and Cryptography
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The Gleason---Pierce---Ward theorem gives constraints on the divisor and field size of a linear divisible code over a finite field whose dimension is half of the code length. This result is a departure point for the study of self-dual codes. In recent years, additive codes have been studied intensively because of their use in additive quantum codes. In this work, we generalize the Gleason---Pierce---Ward theorem on linear codes over GF(q), q = p m , to additive codes over GF(q). The first step of our proof is an application of a generalized upper bound on the dimension of a divisible code determined by its weight spectrum. The bound is proved by Ward for linear codes over GF(q), and is generalized by Liu to any code as long as the MacWilliams identities are satisfied. The trace map and an analogous homomorphism $${x\mapsto x-x^p}$$ on GF(q) are used to complete our proof.