Denote by L3 the check matrix that we have been using to describe the [7,4] Hamming code:
0001111 L3=0 1 1 0 0 1 1
It has among its columns each nonzero triple from F32 exactly once. From this and Lemma 3.1.12, we were able to prove
4.2 Hamming codes and data compression
Hamming codes can also be used for data compression allowing a small amount of distortion (loss of information) by
running the machine backwards.
Choose a generator matrix G for a Hamming code Hamr(q) of length n ove
Proof. Set n = (qr 1)/(q 1), the length of the Hamming code C. As the code has redundancy r, its dimension is n r. As
discussed above, the argument applying Lemma 3.1.12 to prove that dmin(C) = 3 for binary codes goes over to the general
case; so C correc
3.2. ENCODING AND INFORMATION 39 (3.1.16) Problem. Prove that the dual of an MDS codes is also an MDS code.
(3.1.17) Problem. Prove that a binary MDS code of length n is one of cfw_0, the repetition code, the parity check code, or all Fn2 .
3.2 Encoding a
In order to dene codes that we can encode and decode efciently, we add more
structure to the codespace. We shall be mainly interested in linear codes. A linear code of
length n over the eld F is a subspace of Fn. Thus the words of t
(3.1.4) Problem. Prove that, in a linear code over the eld Fq, either all of the codewords begin with 0 or exactly 1/q of the codewords
begin with 0. (You might want rst to consider the binary case.)
(3.1.5) Problem. Let C be an [n,k,d] linear code over t
A consequence of the lemma is that minimum distance for linear codes is much easier to calculate than for arbitrary codes.
One need only survey |C| codewords for weight rather than roughly |C|2 pairs for distance.
Examples. Of course the minimum weight of
3.3 Decoding linear codes
A form of decoding always available is dictionary decoding. In this we make a list of all possible received words and next
to each word write the codeword (or codewords) to which it may be decoded under MLD. In decoding, when a w