Proof. If a,b C|K and t,s K, then ta+sb is in Kn, as all entries are from K, and is in C, since C is
linear over F K. Therefore ta+sb C|K, and the subeld subcode is linear.
Clearly C|K has length n. Since it is contained within the linear code C, we must
.
!
Therefore the coefcient of xn1 in c(x)d(x) is 0. If c(x) = n1 cixi and d(x) = i=0
!n1 j
!
djx , then in general the coefcient of xm in c(x)d(x) is
cidj. In j =0 i+j =m
8.2. CYCLIC GRS CODES AND REED-SOLOMON CODES 109 particular, the two
determinations
Chapter 8
Cyclic Codes
Among the rst codes used practically were the cyclic codes which were gen- erated using shift
registers. It was quickly noticed by Prange that the class of cyclic codes has a rich algebraic
structure, the rst indication that algebra
(8.3.4) Corollary. (1) A binary, narrow-sense, primitive BCH code of designed distance 2 is a cyclic
Hamming code.
(2) A binary, narrow-sense, primitive BCH code of designed distance 3 is a cyclic Hamming code.
Proof. Letn=2m1andK=F2 F2m =F. Letbeaprimiti
.
!
Therefore the coefcient of xn1 in c(x)d(x) is 0. If c(x) = n1 cixi and d(x) = i=0
!n1 j
!
djx , then in general the coefcient of xm in c(x)d(x) is
cidj. In j =0 i+j =m
8.2. CYCLIC GRS CODES AND REED-SOLOMON CODES 109 particular, the two
determinations
he check polynomial of C.
Under some circumstances it is convenient to consider xn 1 to be the
generator polynomial of the cyclic code 0 of length n. Then by the theorem, there is a one-to-one
correspondence between cyclic codes of length n and monic divi
6.1.3 Lengthening and shortening
In lengthening or shortening a code we keep its redundancy xed but vary its length and dimension.
When lengthening a code C we increase the length and add codewords to C. The inverse process of
shortening a code involves t
6.3 Extended generalized Reed-Solomon codes
Let n > 1, and consider n-tuples from the eld F with the following properties: (i) w = (w1,w2,.,wn)
Fn has all its entries wi not 0;
(ii) = (1,2,.,n) Fn and = (1,2,.,n) Fn satisfy
ij ji,foralli=j.
For k > 0 the
Chapter 7
Codes over Subelds
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 eld F as a vector
space over its subeld K. We wil
7.3.3 Perfect codes
Although we will not at present devote much time to perfect codes, we emphasize the speciality of
the Golay codes by reporting
(7.3.4) Theorem. (Tietavainen and Van Lint, 1971.) A perfect e-error- correcting code C of length n
over Fq