Basic concepts: Shortening and puncturing, extension and lengthening.
The parity check for binary linear codes. Extended binary Hamming codes.
Residual codes and Griesmer bound.
So far we have concentrated on constructing codes from scratch, so to speak.
One idea led to the Hamming codes, another to the Reed-Solomon codes. Now we want to start studying recursive mechanisms. The most obvious question is: which codes can be found inside a given code?
LetC be a code [n, k, d]q.Fix a coordinate, sayn.
TheshorteningDofCis defined by
D={(x1, x2, . . . , xn−1)|(x1, x2, . . . , xn−1,0)∈ C}.
That means we consider the subcode ofCconsisting of all codewords that end in 0.Then we throw away this superfluous last coordinate. What can we say about the parameters ofD? ClearlyDis linear. As it is defined by one linear equation, the dimension typically isk−1 (the dimension will bekif all words of C should happen to be 0 at the last coordinate). Also, Dis a subcode of C.It follows that the minimum weight (=minimum distance) ofDcannot be smaller thand.This describes our first recursive construction:
5.1 Theorem. If there is a code [n, k, d]q, then shortening yields a subcode [n−1, k−1, d]q.
Here is another elementary idea. This procedure is called projection in algebra. In the coding community it is known aspuncturingortruncation:
define
81
[n,k,d] q
[n-1,k-1,d] q [n-1,k,d-1] q
lengthening extension
shortening puncturing
FIGURE 5.1: Shortening and puncturing
E={(x1, x2, . . . , xn−1)| there isx∈Fq such that (x1, x2, . . . , xn−1, x)∈ C}.
This means we project the codewords ofConto the firstn−1 coordinates.
The number of codewords ofEis the same as inC (if two different words inC agreed in all but one coordinate, their distance would be 1). In particular,E has dimensionk.This time we cannot expect the minimum distance to remain unchanged. However, as only one coordinate has been removed, the minimum distance will decrease by at most 1.
5.2 Theorem. If there is a code [n, k, d]q, where d > 1, then puncturing (projection) yields a code[n−1, k, d−1]q.
For example, given a code [24,12,8]2(it is known as thebinary extended Golay code), we can construct by repeated shortening and puncturing
[23,12,7]2, [23,11,8]2, [22,12,6]2, [22,11,7]2, [22,10,8]2, . . .
This raises the interesting question when one of these constructions can be reversed. Given a code [n−1, k−1, d]q,can we find a code [n, k, d]q such that the original code is obtained by shortening? This process is calledextension.
Clearly it cannot always be possible. The process of shortening shows that parameters [n, k, d]q are stronger than [n−1, k−1, d]q. It is natural to ask what is the maximum minimum distance of a one-step extension. The covering radius will shed new light on this problem; see Section 14.2.
In the case of puncturing, the situation is analogous. The inverse of punc- turing is called lengthening and is clearly not always possible. More gen- erally, we will speak of (i-step) lengthening of a code C of length n if the new code of lengthn+ihas the same dimension as C and projects toC; we speak of (i-step) extensionif we have a code of length n+i and repeated shortening at someicoordinates producesC.Unfortunately, the terminology in the literature is rather chaotic.
Here is a famous case when lengthening works: letCbe an [n, k, d]2,where dis odd. Append a new coordinate, where the entry is chosen such that the new lengthened codeword has even weight. This means that the entry in the last coordinate is 0 if the weight of the old codeword is even, and the entry in the last coordinate is 1 if the old weight was odd. Clearly the new code is linear and has the same dimension as the old one. Also, in the lengthened code all weights are even. In particular, weight ddoes not occur. We have a code [n+ 1, k, d+ 1]2. The entry in the extra coordinate is known as the parity check bit.
5.3 Theorem. Let C be a binary code[n, k, d]2,wheredis odd. Appending a parity check bit yields a code[n+ 1, k, d+ 1]2.
In particular, a code [n, k, d]2 with oddd exists if and only if a code [n+ 1, k, d+ 1]2 exists. In the theory of binary linear codes one can therefore concentrate on even minimum weight. However, this is a specialty of the binary case.
We have seen the binary Hamming code [7,4,3]2 in Sections 2.1, 2.2 and 2.5. A generator matrix ofH3(2) was given in Section 2.1:
1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 1 0
After addition of a parity check bit, we obtain the matrix
1 0 0 1 1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 1 1 0 1
It is a generator matrix of the so-calledextended Hamming code [8,4,4]2
(alas: we would prefer to call it the lengthened Hamming code). We check that any two rows are orthogonal. This means that the extended Hamming code is orthogonal to itself. As its dimension is just half the length (see Lemma 2.9), we conclude that it equals its dual. Such codes are calledself-dual.
5.4 Theorem. The extended binary Hamming code[8,4,4]2is self-dual (C=C⊥).
We can apply the parity check to all the binary Hamming codes [2r−1,2r− (r+ 1),3]2. This yields the extended (really lengthened) binary Hamming codes:
5.5 Theorem. The extended binary Hamming codeshave parameters [2r,2r−(r+ 1),4]2.
Examples are
[8,4,4]2, [16,11,4]2, [32,26,4]2.
Residual codes, Griesmer bound
LetCbe an [n, k, d]q code andv∈ Ca codeword of weightd.We can assume the nonzero entries of v arev1 =v2 =· · ·=vd = 1. Otherwise permute the coordinates and multiply all codewords by suitable nonzero field elements in the first dcoordinates. This produces a code with the same parameters.
Choose a generator matrix G whose first row is v. The residual code R is the code generated by the remaining k−1 rows ofG, after projection to the last n−d coordinates (where v has zero entries); in other words, R is generated by the matrix in the southeast corner of G.Clearly R is a linear code of length n−d. It has dimensionk−1, as otherwise C would contain a nonzero codeword of weight < d. What can be said about the minimum distanced′ ofR?
Let 0 6=x∈ C be a linear combination of the last k−1 rows of Gwhose image inR(consisting of the lastn−dcoordinates) has weightw.We want a lower bound on w.Letδi be the number of coordinates in the first section wherexhas entryi,for arbitraryi∈Fq.The codewordx−iv∈ C has weight w+d−δi. This weight must be≥d.It follows thatw≥δi. As the average value ofδi isd/q,we havew≥ ⌈d/q⌉,the smallest integer at least as large as d/q.We have seen the following:
5.6 Proposition. Let C be an [n, k, d]q code of precise minimum weight d.
The residual code is an [n−d, k−1,⌈d/q⌉]q code.
Repeated application of Proposition 5.6 yields n ≥ Pk−1
i=0⌈d/qi⌉. This is true even if at any point in the process the true minimal distance should be larger than its guaranteed lower bound.
5.7 Theorem (Griesmer bound). If a linear [n, k, d]q code exists, then n≥
k−1X
i=0
⌈d/qi⌉.
The Griesmer bound is very useful. It is a bound on linear codes. How does it compare to the Singleton bound? The lower bound onn is a sum of k integers. The largest of these integers is d; all are strictly positive. This impliesn≥d+ (k−1).We see that the Griesmer bound implies the Singleton bound Theorem 4.1. The Griesmer bound was proved by Griesmer [99] in the binary case, by Solomon and Stiffler [196] in the generalq-ary case.
As first examples, codes [8,4,4]2, [11,3,6]2and [12,6,6]3meet the Griesmer bound with equality. In Chapter 17 we will encounter an even more natural proof of the Griesmer bound in terms of the geometric description of linear codes; see Exercise 17.1.8.
1. LetC be a linear code [n, k, d]q.
2. One-stepshorteningproduces [n−1, k−1, d]q, one-stepprojectionyields [n−1, k, d−1]q.
3. We speak of a one-step extension if [n+ 1, k+ 1, d]q can be obtained, of a one-steplengtheningif [n+ 1, k, d+ 1]q
can be constructed.
4. Theparity check works only in the binary case.
It constructs [n+ 1, k, d+ 1]2from [n, k, d]2 whendis odd.
5. Theextended binary Hamming codeshave parameters [2r,2r−(r+ 1),4]2.
6. The Griesmer bound: n ≥
k−1X
i=0
⌈d/qi⌉ is stronger than the Singleton bound.
Exercises 5.1
5.1.1. Let Cbe a binary linear code with odd minimum distancedandD ⊂ C the set of codewords of even weight. Prove thatDis a subspace of codimension 1 inC.
5.1.2. Given a linear code [192,11,92]2 (it exists), make a list of all param- eters of codes of lengths188and above that can be constructed using iterated shortening and projection.
5.1.3. Prove the following: if a code is MDS, then every code derived from it by shortening or projection is MDS as well.
5.1.4. Determine the parameters of the three-dimensional binary code gener- ated by
G=
111000000111100 000111000110011 000111111111100
.
Add a parity check bit to this generator matrix. What are the parameters of the resulting extended code?
5.1.5. Let X = {1,2, . . . ,8}. Define a binary matrix M of size (8,24) as follows: the rows of M are indexed by the elements x ∈ X. The columns of M are indexed by unordered pairs {i, j} ⊂ X, where {i, j} is none of {1,2},{3,4},{5,6},{7,8}(these forbidden pairs form what is known as aone- factor). The entry in row xand column {i, j} is0 if x=i orx=j;it is 1 otherwise. Observe that every column of M has weight 6 and every row has weight18.
Prove thatM has rank7 and that its row-space is a code[24,7,10]2. This description is from Jaffe [119].
5.1.6. Given a linear code[24,7,10]2,which parameters of6- or7-dimensional codes can be constructed using iterated shortening and projection?
5.1.7. Consider the matrix
A=
0111111111 1011112222 0001021221 0011220102
from Section 3.6, which generates a code [10,4,6]3. Find generator matrices for the codes derived by shortening and by projection.
5.1.8. Let C be an[n, k, d]q code, x= (x1, . . . , xw,0,0, . . .)∈ C⊥ a codeword of weight w in the dual code. Apply shortening to C with respect to the first w coordinates. We know that the resulting code has parameters (at least) [n−w, k−w, d]q.
Show that this shortened code has in fact better parameters.
This is known asconstruction Y1.
5.1.9. From Section 3.4 we know the Simplex codes. Their duals are the Hamming codes. Determine the parameters of the codes obtained by applying constructionY1 to the Simplex codes.
5.1.10. Show that a code with minimum distanced > q cannot be MDS.
5.1.11. Prove that the Simplex codes meet the Griesmer bound with equality.
5.1.12. Prove the following generalization of the residual code construction, which is used in Dodunekov [72] and Groneick and Grosse [100]:
Let C be an [n, k, d]q code and v ∈ C a codeword of weight w. Assume d−w+⌈w/q⌉>0.
Then there is an [n−w, k−1, d−w+⌈w/q⌉]q code.