Name
Abstract | ||
|---|---|---|
Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), whose nonzero codewords have period equal to the length of the code. Asymptotics are derived on the function
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Gamma (k,q)$ </tex-math></inline-formula>
, that is defined as the largest number of nonzero weights a cyclic code of dimension
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>
over
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>
can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>
-ary, as well as in the
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>
-ary Hamming code are determined, two difficult results of independent interest. |
Year | DOI | Venue |
|---|---|---|
2020 | 10.1109/TIT.2019.2946660 | IEEE Transactions on Information Theory |
Keywords | Field | DocType |
Upper bound,Linear codes,Generators,Orbits,Signal processing,Signal processing algorithms | Discrete mathematics,Hamming code,Upper and lower bounds,Cyclic code,Asymptotic analysis,Mathematics,Alphabet,Binary number | Journal |
Volume | Issue | ISSN |
66 | 3 | 0018-9448 |
Citations | PageRank | References |
2 | 0.38 | 0 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Minjia Shi | 1 | 11 | 4.81 |
| Xiaoxiao Li | 2 | 22 | 11.81 |
| Alessandro Neri | 3 | 14 | 6.10 |
| Patrick Solé | 4 | 636 | 89.68 |