MinT - Extended Reed–Solomon Codes for OOAs

Extended Reed–Solomon Codes for OOAs

For every field F_b, every T ≥ 1, and every 0 ≤ n ≤ bT the (extended) Reed-Solomon NRT-codes [1] are linear [(s, T ), n, sT −n + 1]-NRT-codes with s = b (simple case) and s = b + 1 (extended case). Their duals are linear ordered orthogonal arrays OOA(b^sT−n, s,F_b, T , sT −n).

Construction of the Reed-Solomon NRT-Code

Let T ≥ 1 and let x₁,…, x_b denote the b elements of F_b. The Reed-Solomon NRT-code RS(T ;n, b) for n = 0,…, bT is defined as

RS(T ;n, b) := {(f^{(T −1)}(x_i),…, f⁽⁰⁾(x_i))_{i=1,…, b} : f ∈ F_b[X]withdeg p < n}

where f⁽ⁱ⁾(x) denotes the ith coefficient in the Taylor expansion of f at x. It is a linear [(b, T ), n, bT −n + 1]-code; its dual is a linear OOA(b^bT−n, b,F_b, T , bT −n). Reed-Solomon codes over a given field F_b are all subcodes of each other, i.e., we have RS(T ;n, b) ⊂ RS(T ;n + 1, b).

Reed-Solomon NRT-codes can also be interpreted as algebraic-geometric NRT-codes based on the rational function field F_b(X).

The Extended Reed-Solomon NRT-Code

The extended Reed-Solomon code RS_e(T ;n, b) for n = 0,…, bT is obtained by standard lengthening of RS(T ;n, b). It can also be interpreted as RS(T ;n, b) with an additional factor defined by evaluating f^{(T −1)},…, f⁽⁰⁾ at infinity. RS_e(T ;n, b) is a linear [(b + 1, T ), n,(b + 1)T −n + 1]-code; its dual is a linear OOA(b^(b+1)T−n, b + 1,F_b, T ,(b + 1)T −n).

Optimality

Reed-Solomon codes and extended Reed-Solomon codes meet the Singleton bound for OOAs and NRT-codes with equality and are therefore MDS-NRT-codes. Viewed as OOAs, they are OOAs with index unity.

Special Cases

RS(T ;0, b) is the [(b, T ), 0, bT + 1]-trivial NRT-code.
RS(T ;1, b) has the same parameters as the [(b, T ), 1, bT ]-repetition NRT-code.
RS(T ;bT , b) is the [(b, T ), bT , 1]-NRT-code without redundancy.
RS_e(T ;0, b) is the [(b + 1, T ), 0,(b + 1)T + 1]-trivial NRT-code.
RS_e(T ;1, b) has the same parameters as the [(b + 1, T ), 1,(b + 1)T ]-repetition NRT-code.

References

[1]	Michael Yu. Rosenbloom and Michael A. Tsfasman. Codes for the m-metric. Problems of Information Transmission, 33:55–63, 1997.

Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Extended Reed–Solomon Codes for OOAs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_OReedSolomon-extended.html

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