Singular Manual: Codes and the decoding problem

| ### C.8.1 Codes and the decoding problem #### Codes Let 799#799 be a field with 800#800 elements. A linear code 78#78 is a linear subspace of 801#801 endowed with the Hamming metric. Hamming distance between x,y 802#802. Hamming weight of x 803#803. Minimum distance of the code 804#804. The code 78#78 of dimension 282#282 and minimum distance 171#171 is denoted as 805#805. A matrix 190#190 whose rows are the base vectors of 78#78 is the generator matrix. A matrix 806#806 with the property 807#807 is the check matrix. #### Cyclic codes The code 78#78 is cyclic, if for every codeword 808#808 in 78#78 its cyclic shift 809#809 is again a codeword in 78#78. When working with cyclic codes, vectors are usually presented as polynomials. So 810#810 is represented by the polynomial 811#811 with 812#812, more precisely 813#813 is an element of the factor ring 814#814. Cyclic codes over 799#799 of length 17#17 correspond one-to-one to ideals in this factor ring. We assume for cyclic codes that 815#815. Let 816#816 be the splitting field of 817#817 over 799#799. Then 710#710 has a primitive 17#17-th root of unity which will be denoted by 4#4. A cyclic code is uniquely given by a defining set 818#818 which is a subset of 819#819 such that 820#820 A cyclic code has several defining sets. #### Decoding problem Complete decoding: Given 821#821 and a code 822#822, so that 41#41 is at distance 823#823 from the code, find 824#824. Bounded up to half the minimum distance: With the additional assumption 825#825, a codeword with the above property is unique. #### Decoding via systems solving One distinguishes between two concepts: Generic decoding: Solve some system 826#826 and obtain some "closed" formulas 827#827. Evaluating these formulas at data specific to a received word 828#828 should yield a solution to the decoding problem. For example for 829#829. The roots of 830#830 yield error positions, see the section on the general error-locator polynomial. Online decoding: Solve some system 831#831. The solutions should solve the decoding problem. #### Computational effort Generic decoding. Here, preprocessing is very hard, whereas decoding is relatively simple (if the formulas are sparse). Online decoding. In this case, decoding is the hard part.

User manual for Singular version 4.4.1, 2025, generated by texi2html.