## Codes of Belov Type

For a prime power *b* and positive integers *n* and *d*, write

*d*=

*wb*

^{n−1}–

*q*

^{ui−1}

where *w* = ⌈*d* /*b*^{n−1}⌉, *n* > *u*_{1} ≥ *u*_{2} ≥ ⋯ ≥ *u*_{v} ≥ 1, and at most *b*−1 *u*_{i}’s take any given integer value. Then there exists an [*s*, *n*, *d*]-code C over **F**_{b} meeting the Griesmer bound with equality provided that

*u*

_{i}≤

*wn*.

The special case for *b* = 2 is shown in [1], therefore the resulting codes are usually denoted as codes of Belov type. The general result is from [2].

### Construction

Consider the multiset containing every point of the projective space PG(*n*−1, *b*) exactly *w* times and remove subspaces *U*_{i} of dimension *u*_{i} − 1 for *i* = 1,…, *v*. Obviously, the subspace *U*_{i} must be chosen such that no point of PG(*n*−1, *b*) is contained in more than *w* of them. This can be done as follows: Identify a point (*a*_{0} : … : *a*_{n−1}) ∈ PG(*n*−1, *b*) with the polynomial *a*_{0} + *a*_{1}*X* + … + *a*_{n−1}*X*^{n−1} ∈ **F**_{b}[*X*]. For each *i* = 1,…, *v* choose a different monic, irreducible polynomial *p*_{i} of degree *k* – *u*_{i} (this is always possible, because there are at least *b*−1 such polynomials of each degree in **F**_{b}[*X*]). Then *U*_{i} can be identified with the monic polynomials in ⟨*p*_{i}⟩ with degree less than *n*.

The remaining points form the columns of the generator matrix of C. In other words C is a properly truncated version of a *w* times juxtapositioned simplex code.

### See Also

### References

[1] | B. I. Belov. A conjecture on the Griesmer boundary. In Optimization methods and their applications (All-Union Summer Sem., Khakusy, Lake Baikal, 1972) (Russian), pages 100–106, 182. Sibirsk. Ènerget. Inst. Sibirsk. Otdel. Akad. Nauk SSSR, Irkutsk, 1974. |

[2] | Raymond Hill. Optimal linear codes. In Cryptography and Coding, II (Cirencester, 1989), volume 33 of Inst. Math. Appl. Conf. Ser. New Ser., pages 75–104. Oxford Univ. Press, New York, 1992. |

[3] | F. Jessie MacWilliams and Neil J. A. Sloane.The Theory of Error-Correcting Codes.North-Holland, Amsterdam, 1977. |

[4] | Andries E. Brouwer. Bounds on the size of linear codes. In Vera S. Pless and W. Cary Huffman, editors, Handbook of Coding Theory, volume 1, pages 295–461. Elsevier Science, 1998. |

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Codes of Belov Type.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CBelov.html