Minimal Codes From Characteristic Functions Not Satisfying The Ashikhmin-Barg Condition
Minimal Codes From Characteristic Functions Not Satisfying The Ashikhmin-Barg Condition
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发表时间：
2019-12
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通讯作者：
J. Sorci
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作者：
J. Sorci
最小码是一种线性码，其中一个码字在另一个码字的支持中包含其支持的唯一实例是码字是彼此的标量倍数。Ashikhmin和Barg给出了码最小的充分条件，这引起了人们对构造不满足条件的最小码的兴趣。考虑当$f$是一组点的指示函数时的一类特殊码$\mathcal C_f$，根据$f$支持的某些几何性质，证明了$\mathcal C_f$最小且不满足Ashikhmin和Barg条件的充分条件。我们给出了满足这些几何性质的点集大小的一个下界，并证明了这个下界是紧的。
A minimal code is a linear code where the only instance that a codeword has its support contained in the support of another codeword is when the codewords are scalar multiples of each other. Ashikhmin and Barg gave a sufficient condition for a code to be minimal, which led to much interest in constructing minimal codes that do not satisfy their condition. We consider a particular family of codes $\mathcal C_f$ when $f$ is the indicator function of a set of points, and prove a sufficient condition for $\mathcal C_f$ to be minimal and not satisfy Ashikhmin and Barg's condition based on certain geometric properties of the support of $f$. We give a lower bound on the size of a set of points satisfying these geometric properties and show that the bound is tight.