Evaluation of the Hamming weights of a class of linear codes based on Gauss sums. (English) Zbl 1381.94118
Summary: Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of \(q\)-ary linear codes under some certain conditions, where \(q\) is a power of a prime. The lower bound of its minimum Hamming distance is obtained. In some special cases, we evaluate the weight distributions of the linear codes by semi-primitive Gauss sums and obtain some one-weight, two-weight linear codes. It is quite interesting that we find new optimal codes achieving some bounds on linear codes. The linear codes in this paper can be used in secret sharing schemes, authentication codes and data storage systems.
MSC:
| 94B05 | Linear codes (general theory) |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
| 11T23 | Exponential sums |
| 94A62 | Authentication, digital signatures and secret sharing |
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