Extended Reed–Solomon Codes for OOAs

For every field Fb, every T ≥ 1, and every 0 ≤ nbT 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(bsT−n, s,Fb, T , sT n).

Construction of the Reed-Solomon NRT-Code

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

RS(T ;n, b) := {(f(T −1)(xi),…, f(0)(xi))i=1,…, b  :  fFb[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(bbT−n, b,Fb, T , bT n). Reed-Solomon codes over a given field Fb 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 Fb(X).

The Extended Reed-Solomon NRT-Code

The extended Reed-Solomon code RSe(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. RSe(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,Fb, 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

See Also

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: 2008-04-04. http://mint.sbg.ac.at/desc_OReedSolomon-extended.html

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