Applications of finite geometry in coding theory and cryptography

More documents

Recommendations

Info

Theorem 7 (Segre, Thas [27, 32])For• q an odd prime power,• 2 ≤ k < √ q/4,every [n = q + 1, k, d = q + 2 − k]-MDS code is a GDRS code.This preceding result was obtained using methods from algebraic geometry and projectionarguments.The motivation for the next result is as follows. The GDRS codes are MDS codes oflength q + 1. Maybe they can be extended to MDS codes of length q + 2. The followingresult proves that this is practically never the case.Theorem 8 (Storme [31])Consider the [q + 1, k, q − k + 2] q -GDRS code.For q odd and 2 ≤ k ≤ q + 3 − 6 √ q log q, and for q even and 4 ≤ k ≤ q + 3 −6 √ q log q, this [q+1, k, q+2−k] q -GDRS code cannot be not extended to a [q+2, k, q+3 − k] q -MDS code.2.3. Minihypers and the Griesmer boundLet N q (d, k) denote the minimal n for which an [n, k, d] q -code exists and let ⌈x⌉ denotethe smallest integer larger than or equal to x.Theorem 9 (Griesmer bound [9, 30])N q (k, d) ≥ d + N q (k − 1, ⌈ d ⌉) (1)qandk−1∑⌈ ⌉ dN q (k, d) ≥ G q (k, d) =q i . (2)i=0Proof. Let C be an [n, k, d] q -code. Without loss of generality we can assume that Ccontains the codeword (0, . . .,0, 1, . . .,1) of weight d.Thus we have a generator matrix of the formG =( )0 · · · 0 1 · · · 1.G 1 G 2This matrix G 1 has rank k − 1 since otherwise we could make a row of G 1 zero andC would contain a codeword of weight less than d. Thus G 1 is the generator matrix ofan [n − d, k − 1, d 1 ] q -code.Let (u, v) ∈ C, with w(u) = d 1 . Since also all codewords of the form (u, v + a1)are in C, we can select v with weight at most ⌊ q−1q d⌋.Since (u, v) ∈ C, we have w(u) + w(v) ≥ d or d 1 ≥ d − ⌈ d q⌉. This provesEquation (1).
Iterating Equation (1) gives:N q (k, d) ≥ d + N q (k − 1, ⌈ d q ⌉)≥ d + ⌈ d q ⌉ + N q(k − 2, ⌈ d q 2 ⌉).k−2∑⌈ ⌉ d≥q ii=0k−1∑⌈ ⌉ d≥q i .i=0+ N q (1, ⌈ d ⌉)qk−1 Now we want to construct linear codes that meet the Griesmer bound, i.e. we areinterested in [G q (k, d), k, d] q -codes.By θ k = qk −1q−1 , we denote the number of 1-dimensional subspaces of Fk q , i.e. thenumber of points in PG(k − 1, q).The simplex code S k is a [θ k , k, q k−1 ] q -code whose generator matrix is formed byθ k pairwise linearly independent vectors in F k q. For each t, the copy of t simplex codesis a [tθ k , tk, tq k−1 ] q -code that satisfies the Griesmer bound.An excellent way to construct more linear codes satisfying the Griesmer bound isto start with a copy of t simplex codes and delete columns of the generator matrix. Thecolumns to be deleted form the generator matrix of what is called an anticode. This isa code with an upper bound on the distance between its codewords. Even the distance 0between codewords is allowed, i.e. an anticode may contain repeated codewords.Definition 3If G is a k × m matrix of F q , then the q k combinations of its rows form the codewordsof an anticode of length m. The maximum distance δ of the anticode is the maximumweight of any of its codewords. If rank G = r, each codeword occurs q k−r times.If we start with t copies of the simplex code and delete m columns that form ananticode with maximum distance δ, we obtain a [tθ k − m, k, tq k−1 − δ] q -code.Codes meeting the Griesmer bound and their anticodes have a nice geometrical interpretation.Let C be an [n, k] q -code with generator matrix G. Each column of the generatormatrix describes a point of PG(k − 1, q). We represent C by the multiset M of these npoints.For instance, the simplex code S k is represented by the point set of PG(k − 1, q).An i-point is a point of multiplicity i. For each subset S of PG(k − 1, q), we denotethe number of points of M in S by c(S). Letγ i = max{c(S) | S is a subspace of dimension i} .□
Page 1 and 2: Applications of finite geometry in
Page 3 and 4: £A 1¡A 2¦Each line 〈v, w〉 of
Page 5 and 6: As indicated in the abstract, proje
Page 7: ⎛⎞1 0 g 1,k+1 . . . g 1,n⎜G =
Page 11 and 12: θ k+1 − θ k−s+1θ s+ θ k−s
Page 13 and 14: Take two skew planes π and ¯π. L
Page 15 and 16: for every B ⊃ A.Example 8 shows h
Page 17 and 18: 67 67 1 115
Page 19: 3.3. AESIn 1997 the American Nation

Applications of finite geometry in coding theory and cryptography

Create successful ePaper yourself

Delete template?

Save as template?