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® x-xp{xmapsto x-x^p} on GF(q) are used to complete our proof.
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机译：格里森-皮尔斯-沃德定理对线性可分代码的除数和字段大小给出了限制，该限制是在尺寸为代码长度一半的有限域上进行的。这个结果是研究自我对偶密码的出发点。近年来，由于将加性码用于加性量子码中，因此对其进行了深入的研究。在这项工作中，我们将关于GF（q）上的线性代码（q = p m ）的Gleason-Pierce-Ward定理推广为GF（q）上的加性代码。我们证明的第一步是在由其权重频谱确定的可分码的尺寸上应用广义上限。 Ward通过GF（q）上的线性代码证明了边界，只要满足MacWilliams身份，Liu将其推广到任何代码。跟踪图和GF（q）上的类似同质x®x-x p {xmapsto x-x ^ p}用于完善我们的证明。
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