2.5 Coding Theory
2.5.2 Code Products
For an arbitrary field K, the spaceKn is, with the coordinatewise product, a
commutative unitary K-algebra: for all x = (x1, . . . , xn), y = (y1, . . . , yn) ∈
Kn, we define
xy:= (x1y1, . . . , xnyn).
Its unit element is the all-one vector, denoted by 1. The multiplicative group of its invertible elements is (Kn)×= (K×)n, meaning thatx∈Kn is invertible
if and only if all entries of x are non-zero. Given x ∈ (Kn)×, its inverse is
denoted by x−1.
We use this notion of product to multiply spaces as well. Given two vector spacesV, W ⊆Kn, we define their productV W to be theK-linear span of the
set of all productsxy, withx∈V andy∈W. Note that, in general, this set 1I.e. for any pair of triples (α
1, α2, α3),(β1, β2, β3)∈F3 there existsσ∈ GL2(F) such thatσ(αi) =βifori= 1,2,3.
is not additively closed, hence it is strictly contained in its span. Likewise we shall denote the square of a spaceV byV2. Context should prevent confusion with cartesian products.
We will be particularly interested in products of codes, i.e. in the case of a finite base field F. The product of two codesC, D⊆Fn, sometimes called the
Schur product, has usually been denoted byC∗D, but we shall drop the star symbol to lighten notation. For an exhaustive discussion on why our study of code products is well-motivated, the reader is referred to Chapter 1. We can immediately state a trivial upper bound for the product dimension.
Theorem 2.5.9. Let C, D⊆Fn be two codes. Then
dimCD≤dimCdimD and dimC2≤ dimC(dimC+ 1)
2 .
Proof. Let{x1, . . . , xk} and {y1, . . . , y`} be bases ofC and D respectively,
where we set k := dimC and ` := dimD. Then the products xiyj with
1 ≤i≤kand 1≤j ≤`span CDand the products xixj with 1≤i≤j ≤k
spanC2.
In fact it holds that these bounds are achieved by most codes. This is shown in Chapter 3 for the second inequality, while for the first inequality the reader is referred to [64].
The following simple observation will be freely used later.
Lemma 2.5.10. Let x ∈ (Kn)×. For any vector space V ⊆ Kn, we have
dimV = dimxV.
Lemma 2.5.11 below classifies all subalgebras of Kn. For alli = 1, . . . , n, let
ei denote thei-th unit vector inKn. We call a vector of the formPi∈Iei for
someI⊆ {1, . . . , n}aprojector2. In particular,1is the projector with support
{1, . . . , n}. A family of projectors is disjointif the projectors have pairwise disjoint supports.
Lemma 2.5.11. Any K-subalgebra of Kn admits a K-basis of disjoint projec-
tors.
Proof. LetA⊆Knbe aK-subalgebra. We argue by induction onk:= dimA.
Ifk= 1 thenAis generated by a vectorxwhose non-zero coordinates must be all equal, otherwisex2 is not a
K-multiple ofx. Ifk >1, pickx∈A, x6= 0 of
minimal support with one of its coordinates equal to 1, and let{x1=x, . . . , xk}
2To justify our terminology, note that ifV ⊆
Kn is a vector space and andx∈Knis a projector, thenxV can be viewed as the projection ofV onto the support ofx.
be a K-basis of Acontainingx. Then xis a projector, otherwise x2−x6= 0
would have smaller support. For alli= 2, . . . , k, if suppxiand suppxintersect,
say in position j, then we can choose λi ∈ K so that xi +λix has a zero
in position j, hence x(xi+λix) ∈ A has support strictly smaller than x.
By minimality of suppx, x(xi+λix) = 0, i.e. xand xi+λixhave disjoint
support. Replacing if need bexibyxi+λix, we obtain thatAis a direct sum
A =hxi ⊕ hx2, . . . , xki and the conclusion follows by applying the induction
hypothesis to the second summand.
Remark 2.5.12. Lemma 2.5.11 implies in particular that the number of sub- algebras of Kn is finite.
LetV ⊆Kn be aK-vector space. We define St(V) :={x∈Kn :xV ⊆V}, the
stabilizerofV inKn. AsV is linear, St(V) is aK-algebra, hence Lemma 2.5.11
applies. In particular, as St(V) has a basis of vectors whose entries are all 0’s and 1’s, it is invariant under base-field extension3, i.e. the following lemma holds.
Lemma 2.5.13. Let K0/K be a field extension. Let V ⊆ Kn be a K-vector
space. Then
St(V ⊗K0) = St(V)⊗ K0.
LetC⊆Fn be a code. As in the vector-space case, we can define its stabilizer
and apply Lemma 2.5.11, which yields an F-basis {π1, . . . , πh} of St(C) of
disjoint projectors, where h := dim St(C). When h = 1 we say that C has trivial stabilizer, or that it is indecomposable. We have the following lemma, whose proof is straightforward.
Lemma 2.5.14. Any full-support code C⊆Fn decomposes as C=π1C⊕ · · · ⊕πhC
where {π1, . . . , πh} is an F-basis of disjoint projectors of St(C). Moreover,
each summand πiC, viewed as a code in Fwt(πi), is indecomposable and has
full support.
Facts on stabilizers, including Lemmas 2.5.13 and 2.5.14, can be found in [65, from §2.6 onwards].
Lemma 2.5.14 states in particular that a full-support code has non-trivial sta- bilizer if and only if it decomposes as a direct sum of codes, and the dimension of the stabilizer equals the number of indecomposable components. It follows that all MDS codes, except the trivial codeFn, have trivial stabilizer.
3If
K0/Kis a field extension, the base-field extensionV ⊗K0, where the tensor product is taken overK, ofV is theK0-span ofV.
We continue this section with two refinements of the classical Singleton Bound, involving the dimension of St(C) beside the usual parameters. They naturally reduce to the classical Singleton Bound when the code C is indecomposable, i.e. dim St(C) = 1.
Lemma 2.5.15. LetC⊆Fn be a code.
1. If dmin(C)>1 then
dmin(C)≤n−dimC+ 1−(dim St(C)−1).
2. If C has full support then
dmin(C)≤
n−dimC
dim St(C)+ 1.
Proof. We may assume that C has full support, as the first claim in the general case follows immediately from the first claim in the full-support case. Set k := dimC, d := dmin(C) and h := dim St(C). By Lemma 2.5.14 we
have that C is a direct sumC =C1⊕ · · · ⊕Ch of full-support codes. For all
i = 1, . . . , h, letni, ki and di denote the support size, the dimension and the
minimum distance of Ci respectively. We havePhi=1ni=n,Phi=1ki =k and
d= min
i {di} ≤mini {ni−ki}+ 1
by the classical Singleton Bound. In the cased >1 we haveni−ki≥di−1≥1
for alli= 1, . . . , h, hence, for all j= 1, . . . , h,
nj−kj =n−k−
X
i6=j
(ni−ki)≤n−k−(h−1).
Putting everything together we have
d= min
i {di} ≤mini {ni−ki}+ 1≤n−k+ 1−(h−1),
which proves the first claim. To prove the second claim, note that
n−k= h X i=1 (ni−ki)≥hmin i {ni−ki},
hence mini{ni−ki} ≤(n−k)/hand the conclusion follows.
We conclude this section with some remarks on the effect of the product opera- tion on MDS codes. The two results below relate the dimension of the product of two codes with the MDS property.
Theorem 2.5.16. Let C, D ⊆ Fn be full-support codes. If (at least) one of
them is MDS, then
dimCD≥min{n,dimC+ dimD−1}.
Proof. See [65,§3.5].
Lemma 2.5.17. Let C, D⊆Fn be MDS codes such that
dimCD= dimC+ dimD−1.
ThenCD is MDS.
Proof. By Lemma 2.5.5, it suffices to show that for any choice of I ⊆ {1, . . . , n} with |I| =d∗ := n+ 1−dimCD there exist x∈ C, y ∈ D with suppxy = I. Without loss of generality, assume that I = {1, . . . , d∗}. As
C and D are both MDS, there exist x ∈ C and y ∈ D such that suppx =
I∪ {d∗+ 1, . . . , dC}and suppy=I∪ {n−(dD−d∗) + 1, . . . , n}, wheredCand
dD denote the minimum distance of C andD respectively. One checks that
dC=n−(dD−d∗), hence indeed suppxy= suppx∩suppy=I.
Finally, we observe that if CandD are two Reed-Solomon codes with a com- mon evaluation-point sequenceα, then the productCD is also Reed-Solomon with evaluation-point sequence α and we have dimCD = min{n,dimC + dimD−1} which, as Theorem 2.5.16 states, is the minimum possible dimen- sion of the product of MDS codes.