Arts & Sciences Events
[PAST EVENT] Mathematics Colloquium - Shuxing Li (Univ of Delaware)
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- Free food
- Open to the public
Counting Fearlessly and Meticulously: Weight Distribution of Cyclic Codes with Generalized Niho-type Nonzeros
Determining the minimum Hamming distance of an error-correcting code C has long stood as a fundamental challenge in coding theory. In this talk, we turn our attention to an even more ambitious problem: determining the full weight distribution of C. This means counting the number of codewords in C of each possible Hamming weight—an essential but notoriously difficult task, especially for general codes.
While computing the weight distribution for an arbitrary code remains largely out of reach, the problem becomes more tractable under additional structural assumptions. Specifically, if the code C is cyclic, one can follow a well-established approach to reduce the problem to the number-theoretic task of evaluating certain exponential sums.
In this talk, we focus on a family of cyclic codes characterized by generalized Niho-type nonzeroes. For these codes, we succeed in determining the value distribution of the associated exponential sums through a process of fearless and meticulous counting. Our approach draws upon a rich toolkit of mathematical ingredients, including exponential sums, elementary symmetric polynomials, binary Zetterberg codes, and the analysis of Vandermonde-like homogeneous linear systems.
Sponsored by: Mathematics
Contact
Eric Swartz