可除码 (Divisible Codes)

A linear code over $\mathbb{F}_q$ with the Hamming metric is called $Δ$-divisible if the weights of all codewords are divisible by $Δ$. They have been introduced by Harold Ward a few decades ago. Applications include subspace codes, partial spreads, vector space partitions, and distance optimal codes. The determination of the possible lengths of projective divisible codes is an interesting and comprehensive challenge.

翻译：在汉明度量下，$\mathbb{F}_q$上的线性码若其所有码字的重量均能被$Δ$整除，则称为$Δ$-可除码。这类码由Harold Ward在数十年前引入。其应用包括子空间码、部分展形、向量空间划分以及距离最优码。确定射影可除码的可能长度是一个有趣且具有广泛意义的挑战。