Soft decision decoding via sphero-ellipsoidal coverings

Previously, general random-search algorithms were considered [2], [6] for minimum distance (MD) decoding, which finds the codeword(s) c ∈ C with the minimum Hamming distance d(c, a) to the received vector a. Then the following results are used to reduce the complexity of MD decoding. Evseev has proved [5] that for any linear code C the decoding error probability of MD decoding is at most doubled if we correct only D or fewer errors instead of performing full MD decoding. Another important result by Blinovskii [1] shows that virtually all long linear (n, k) codes have covering radius ρ = D + O(ln n) as n →∞. So, explicit MD decoding needs to test only an order of 2n−k lightest error patterns. To correct any error pattern of weight D, we then need to find any set of n − k positions that covers all D erroneous positions of the received vector a and re-encode the remaining k symbols. More formally, we need to find a covering set T (n, n−k, D), that is a set of vectors e of weight n−k such that any vector of weight D is necessarily covered by at least one vector e ∈ T. The lower and upper bounds on the minimal size |Tmin| are well known [4]: (D) / ( n−k D ) ≤ |Tmin(n, n− k, D)| ≤ [ln ( n−k D ) + 1] (D) / ( n−k D ) . (2)