## Extended Reedâ€“Solomon Codes for OOAs

For every field **F**_{b}, every *T * â‰¥ 1, and every 0 â‰¤ *n* â‰¤ *bT * the (extended) Reed-Solomon NRT-codes [1] are linear [(*s*, *T *), *n*, *sT *âˆ’*n* + 1]-NRT-codes with *s* = *b* (simple case) and *s* = *b* + 1 (extended case). Their duals are linear ordered orthogonal arrays OOA(*b*^{sTâˆ’n}, *s*,**F**_{b}, *T *, *sT *âˆ’*n*).

### Construction of the Reed-Solomon NRT-Code

Let *T * â‰¥ 1 and let *x*_{1},â€¦, *x*_{b} denote the *b* elements of **F**_{b}. The Reed-Solomon NRT-code RS(*T *;*n*, *b*) for *n* = 0,â€¦, *bT * is defined as

*T*;

*n*,

*b*) := {(

*f*

^{(T âˆ’1)}(

*x*

_{i}),â€¦,

*f*

^{(0)}(

*x*

_{i}))

_{i=1,â€¦, b}Â : Â

*f*âˆˆ

**F**

_{b}[

*X*]withdegÂ

*p*<

*n*}

where *f*^{(i)}(*x*) denotes the *i*th coefficient in the Taylor expansion of *f* at *x*. It is a linear [(*b*, *T *), *n*, *bT *âˆ’*n* + 1]-code; its dual is a linear OOA(*b*^{bTâˆ’n}, *b*,**F**_{b}, *T *, *bT *âˆ’*n*). Reed-Solomon codes over a given field **F**_{b} are all subcodes of each other, i.e., we have RS(*T *;*n*, *b*) âŠ‚ 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,â€¦, *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 âˆ’1)},â€¦, *f*^{(0)} at infinity. RS_{e}(*T *;*n*, *b*) is a linear [(*b* + 1, *T *), *n*,(*b* + 1)*T *âˆ’*n* + 1]-code; its dual is a linear OOA(*b*^{(b+1)Tâˆ’n}, *b* + 1,**F**_{b}, *T *,(*b* + 1)*T *âˆ’*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.

### See Also

For

*T*= 1 the common Reed-Solomon code is obtained.Reed-Solomon NRT-codes can be generalized to algebraic-geometric NRT-codes by allowing a more general class of functions instead of polynomials.

### References

[1] | MichaelÂ Yu. Rosenbloom and MichaelÂ A. Tsfasman. Codes for the m-metric.Problems of Information Transmission, 33:55â€“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 Reedâ€“Solomon Codes for OOAs.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2024-09-05.
http://mint.sbg.ac.at/desc_OReedSolomon-extended.html