Subfields - 88 Chapter 7 Codes over Subfields In Chapter 6...

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Chapter 7 Codes over Subfields In Chapter 6 we looked at various general methods for constructing new codes from old codes. Here we concentrate on two more specialized techniques that result from writing the field F as a vector space over its subfield K . We will start with linear codes over F and finish with linear codes over K . Of particular practical importance is the case with K = F 2 . Our work on generalized Reed- Solomon codes over F has given us many powerful codes, but by Theorem 5.1.1 their length is bounded by | F | . Binary generalized Reed-Solomon codes are rendered trivial. 7.1 Basics Let dim K ( F ) = m , and choose e 1 , . . . e m to be a K -basis for F . We define the map φ : F -→ K m given by φ ( α ) = ( a 1 , . . . a m ) where α = a 1 e 1 + · · · + a m e m . For brevity, we shall write ˆ α for the 1 × m row vector φ ( α ) and ˇ α for its transpose φ ( α ) > = ( a 1 , . . . a m ) > , an m × 1 column vector. We extend this notation to any p × q matrix A F p,q , with i, j entry a i,j by letting b A K p,mq be the matrix ˆ a 1 , 1 ˆ a 1 , 2 · · · ˆ a 1 ,j · · · ˆ a 1 ,q ˆ a 2 , 1 ˆ a 2 , 2 · · · ˆ a 2 ,j · · · ˆ a 1 ,q . . . . . . . . . . . . . . . . . . ˆ a i, 1 ˆ a i, 2 · · · ˆ a i,j · · · ˆ a 1 ,q . . . . . . . . . . . . . . . . . . ˆ a p, 1 ˆ a p, 2 · · · ˆ a p,j · · · ˆ a p,q 89
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90 CHAPTER 7. CODES OVER SUBFIELDS and ˇ A K mp,q be the matrix ˇ a 1 , 1 ˇ a 1 , 2 · · · ˇ a 1 ,j · · · ˇ a 1 ,q ˇ a 2 , 1 ˇ a 2 , 2 · · · ˇ a 2 ,j · · · ˇ a 1 ,q . . . . . . . . . . . . . . . . . . ˇ a i, 1 ˇ a i, 2 · · · ˇ a i,j · · · ˇ a 1 ,q . . . . . . . . . . . . . . . . . . ˇ a p, 1 ˇ a p, 2 · · · ˇ a p,j · · · ˇ a p,q For our constructions, A might be a spanning or control matrix for a linear code over F . Then the matrices b A and ˇ A can be thought of as spanning or control matrices for linear codes over K . It must be emphasized that these maps are highly dependent upon the choice of the initial map φ , even though φ has been suppressed in the notation. We shall see below that a careful choice of φ can be of great help. (The general situation in which φ is an arbitrary injection of F into K p , for some field K and some p , is of further interest. Here we will be concerned with the linear case, but see the Problem 7.2.3 below.) 7.2 Expanded codes If C is a code in F n , then the code b C = { ˆ c | c C } in K mn is called an expanded code . expanded code ( 7.2.1) Theorem. If C is an [ n, k, d ] code, then b C is an [ mn, mk, d ] code. Proof. The map x 7→ ˆ x (induced by φ ) is one-to-one and has \ r a + s b = r ˆ a + s ˆ b , for all r, s K and a , b F n . Thus b C is a linear code over K with | b C | = | C | = | F | k = ( | K | m ) k = | K | mk , hence b C has K -dimension mk . (This counting argument is a cheat unless F is finite, the case of greatest interest to us. Instead, one should construct a K -basis for b C out of that for F and an F -basis for C . We do this below, constructing a generator matrix for b C from one for C .) If the coordinate c i of c is nonzero, then ˆ c i is not the zero vector of K m .
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