Berlekamp–Massey algorithm: Difference between revisions

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==Description of algorithm==

The Berlekamp–Massey algorithm is an alternative to the [[Reed–Solomon error correction#Peterson–Gorenstein–Zierler decoder|Reed–Solomon Peterson decoder]] for solving the set of linear equations. It can be summarized as finding the coefficients Λ_''j'' of a polynomial Λ(''x'') so that for all positions ''i'' in an input stream ''S'':

:<math> S_{i + \nu} + \Lambda_1 S_{i+\nu-1} + \cdots + \Lambda_{\nu-1} S_{i+1} + \Lambda_{\nu} S_i = 0. </math>

In the code examples below, ''C''(''x'') is a potential instance of ''Λ''(''x''). The error locator polynomial ''C''(''x'') for ''L'' errors is defined as:

:<math> C(x) = C_{L}C_L \ x^{L} + C_{L-1} \ x^{L-1} + \cdots + C_2 \ x^2 + C_1 \ x + 1 </math>

or reversed:

:<math> C(x) = 1 + C_1 \ x + C_2 \ x^2 + \cdots + C_{L-1} \ x^{L-1} + C_{L}C_L \ x^{L}. </math>

The goal of the algorithm is to determine the minimal degree ''L'' and ''C''(''x'') which results in all [[Decoding methods#Syndrome decoding|syndromes]]

:<math> S_{n}S_n + C_1 \ S_{n-1} + \cdots + C_L \ S_{n-L}</math>

being equal to 0:

:<math> S_{n}S_n + C_1 \ S_{n-1} + \cdots + C_L \ S_{n-L} = 0,\qquad L\le n\le N-1.</math>

Algorithm:

''C''(''x'') is initialized to 1, ''L'' is the current number of assumed errors, and initialized to zero. ''N'' is the total number of syndromes. ''n'' is used as the main iterator and to index the syndromes from 0 to (''N''-1)−1. ''B''(''x'') is a copy of the last ''C''(''x'') since ''L'' was updated and initialized to 1. ''b'' is a copy of the last discrepancy ''d'' (explained below) since ''L'' was updated and initialized to 1. ''m'' is the number of iterations since ''L'', ''B''(''x''), and ''b'' were updated and initialized to 1.

Each iteration of the algorithm calculates a discrepancy ''d''. At iteration ''k'' this would be:

:<math> d \gets S_{k}S_k + C_1 \ S_{k-1} + \cdots + C_L \ S_{k-L}.</math>

If ''d'' is zero, the algorithm assumes that ''C''(''x'') and ''L'' are correct for the moment, increments ''m'', and continues.

If ''d'' is not zero, the algorithm adjusts ''C''(''x'') so that a recalculation of ''d'' would be zero:

:<math>C(x) \gets C(x) \ - \ (d / b) \ x^m \ B(x).</math>

The ''x^m'' term ''shifts'' B(x) so it follows the syndromes corresponding to ''b''. If the previous update of ''L'' occurred on iteration ''j'', then ''m'' = ''k'' -− ''j'', and a recalculated discrepancy would be:

:<math> d \gets S_{k}S_k + C_1 \ S_{k-1} + \cdots - (d/b) (S_{j}S_j + B_1 \ S_{j-1} + \cdots ).</math>

This would change a recalculated discrepancy to:

:<math> d = d - (d/b)b = d - d = 0. \ </math>

The algorithm also needs to increase ''L'' (number of errors) as needed. If ''L'' equals the actual number of errors, then during the iteration process, the discrepancies will become zero before ''n'' becomes greater than or equal to (2 ''L''). Otherwise ''L'' is updated and algorithm will update ''B''(''x''), ''b'', increase ''L'', and reset ''m'' = 1. The formula ''L'' = (''n'' + 1 -− ''L'') limits ''L'' to the number of available syndromes used to calculate discrepancies, and also handles the case where ''L'' increases by more than 1.

==Code sample==