| ### C.8.5 Decoding method based on quadratic equations #### Preliminary definitions Let 943#943 be a basis of 801#801 and let 196#196 be the 236#236 matrix with 943#943 as rows. The unknown syndrome 944#944 of a word 836#836 w.r.t 196#196 is the column vector 945#945 with entries 946#946 for 947#947. For two vectors 948#948 define 949#949. Then 950#950 is a linear combination of 943#943, so there are constants 951#951 such that 952#952 The elements 951#951 are the structure constants of the basis 943#943. Let 953#953 be the 954#954 matrix with 955#955 as rows (956#956). Then 943#943 is an ordered MDS basis and 196#196 an MDS matrix if all the 957#957 submatrices of 953#953 have rank 178#178 for all 958#958. #### Expressing known syndromes Let 78#78 be an 799#799-linear code with parameters 805#805. W.l.o.g 959#959. 806#806 is a check matrix of 78#78. Let 960#960 be the rows of 806#806. One can express 961#961 with some 962#962. In other words 963#963 where 191#191 is the 964#964 matrix with entries 965#965. Let 834#834 be a received word with 835#835 and 836#836 an error vector. The syndromes of 828#828 and 836#836 w.r.t 806#806 are equal and known: 966#966 They can be expressed in the unknown syndromes of 836#836 w.r.t 196#196: 967#967 since 961#961 and 968#968. #### Constructing the system Let 196#196 be an MDS matrix with structure constants 969#969. Define 970#970 in the variables 971#971 by 972#972 The ideal 973#973 in 974#974 is generated by 975#975 The ideal 976#976 in 977#977 is generated by 978#978 Let 979#979 be the ideal in 977#977 generated by 973#973 and 976#976. #### Main theorem Let 196#196 be an MDS matrix with structure constants 969#969. Let 806#806 be a check matrix of the code 78#78 such that 963#963 as above. Let 834#834 be a received word with 835#835 the codeword sent and 836#836 the error vector. Suppose that 980#980 and 981#981. Let 503#503 be the smallest positive integer such that 979#979 has a solution 982#982 over the algebraic closure of 799#799. Then 983#983 and the solution is unique and of multiplicity one satisfying 984#984. the reduced Gröbner basis 190#190 for the ideal 979#979 w.r.t any monomial ordering is 985#985 986#986 where 987#987 is the unique solution. For an example see `sysQE` in decodegb_lib. More on this method can be found in [BP2008a].

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