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.

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