Conceptos Básicos
Reconstruction algorithm for Reed-Solomon codes with efficient decoding capabilities.
Resumen
I. Introduction to Sequence Reconstruction Problem by Levenshtein:
Unique reconstruction challenge in noisy channels.
Minimum number of channels required for unique reconstruction.
II. Efficient Decoders:
Majority-logic-with-threshold decoder and its optimality.
Brute-force decoder and list-decoding using a single read.
III. Focus on Reed-Solomon Codes:
Definition and properties of RS codes.
Efficient encoding, decoding, and error correction capabilities.
IV. Soft Decoding by Koetter and Vardy:
Multiplicity matrix construction for soft-decision decoding.
V. Multiplicity Matrix Construction:
Algorithm description for constructing the multiplicity matrix.
Cost analysis of the multiplicity matrix based on reads' distances.
VI. Reconstruction Decoder Using Two Reads:
Algorithm description for reconstruction using two reads at distance ℓ.
Proposition on the correctness and time complexity of the algorithm.
Estadísticas
Nn,q(t, d) ≜ maxc1,c2∈C |Bt(c1) ∩ Bt(c2)|,
Lemma 1 ([1]). Let e ≜ d−1 2 and t ≜ e + ℓ. Then Nn,q(t, d) = t−⌈ d 2⌉ X i=0 n − d i (q − 1)i· X a=d−t+i t−i X b=d−t+i d ad − a b (q − 2)d−a−b !.