Applications of finite geometry in coding theory and cryptography

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Then γ 0 is the maximal i for which an i-point in M exists. The minimum distanceof C is the minimal number of points of M lying in the complement of a hyperplane, i.e.d = n − γ k−2 .If an [n, k, d] q -code meets the Griesmer bound we can compute the values γ i fromits parameters. At this moment we only need the following lemma.Lemma 1 (Maruta [22])Let (s − 1)q k−1 < d ≤ sq k−1 and let C be an [n, k, d] q -code meeting the Griesmerbound. Then γ 0 = max{c(P) | P ∈ PG(k − 1, q)} = s.Proof. By the pigeonhole principle, we get γ 0 ≥nθ k−1> s − 1.Assume γ 0 > s, then there exists a point P = (p 0 , . . .,p k−1 ) described by at leasts + 1 columns of the generator matrix. Consider the subcode C ′ of C defined byk−1C ′ = {x = (x 0 , . . .,x k−1 ) ∈ F k q | ∑x i p i = 0}G .The codewords of C ′ have entry 0 at the columns corresponding to P . Puncturing C ′ atthese columns yields an [n ′ , k ′ , d ′ ] q -code with n ′ ≤ n − s − 1, k ′ = k − 1 and d ′ ≥ d.But the Griesmer bound says that∑n − s − 1 ≥ n ′ ≥ ⌈ d′k−2q i ⌉ ≥ ∑⌈ d k−1q i ⌉ = ∑⌈ d q i ⌉ − ⌈ d ⌉ = n − s,qk−1 a contradiction.k ′i=0i=0We represent the linear code C by the multiset M ′ in which each point P of PG(k −1, q) has weight w(P) equal to s minus the number of columns in the generator matrixdefining P . In fact, M ′ is the multiset of columns of the anticode corresponding to Cin the copy of s simplex codes. We have shown above that for linear codes meeting theGriesmer bound w(P) ≥ 0 for each point P . Let d = sq k−1 − ∑ k−2i=0 t iq i , 0 ≤ t i ≤ q−1for i = 0, . . .,k − 2. Then the total weight of all points in M ′ is ∑ k−2i=0 t iθ i+1 and eachhyperplane has a weight of at least n − d = ∑ k−2i=0 t iθ i .This geometrical structure is important enough to deserve a name.Definition 4An (n, w; d, q)-minihyper is a multiset of n points in PG(d, q) with the property thatevery hyperplane meets it in at least w points.Many characterisation theorems of minihypers are known. The simplest is:Theorem 10 (Bose and Burton [2])Let k ≤ d. A (θ k+1 , θ k ; d, q)-minihyper always is a k-dimensional subspace of PG(d, q).Proof. Let H be a (θ k+1 , θ k ; d, q)-minihyper. We claim that for s ≤ k every codimensions space of PG(d, q) meets H in at least θ k−s+1 points.For s = 1 this is the definition of a minihyper. Now let s > 1 and assume that acodimension s space π meets H in less than θ k−s+1 points. Then the average number ofpoints of H in a codimension s − 1 space through π is less thani=0i=0□
θ k+1 − θ k−s+1θ s+ θ k−s+1 = q k−s+1 + θ k−s+1 = θ k−s+2 ,in contradiction to the already proved result that a codimension s − 1 space contains atleast θ k−s+2 points of H.Now assume that H is not a k-space of PG(d, q), i.e. there is a line l that containsat least two points of H but does not lie completely in H. Let P ∈ l\H. There existsa subspace π ′ of dimension d − k − 1 through P that has no point in common with H.(There are simply not enough points in H to block all the (d −k −1)-spaces through P ).The average number of points of H in a (d−k)-space through π ′ is θ k+1 /θ k+1 = 1.But the (d − k)-space containing l contains at least 2 points of H, thus there must bea (d − k)-space through π ′ that contains no point of H. A contradiction, i.e. H is asubspace.□There are many other characterization results on minihypers. We refer to the literaturefor the known results. As a concrete example of a deep characterization result, wemention the following result of Hamada, Helleseth, and Maekawa.Theorem 11 (Hamada, Helleseth, and Maekawa [10, 11])Let F be a ( ∑ k−2i=0 ɛ iθ i+1 , ∑ k−2i=0 ɛ iθ i ; k −1, q)-minihyper, with ∑ k−2i=0 ɛ i < √ q +1, thenF is the union of ɛ 0 points, ɛ 1 lines, . . ., ɛ k−2 (k − 2)-dimensional subspaces, which allare pairwise disjoint.2.4. Covering radiusFor an e-error correcting code, we search for a large set of pairwise disjoint spheresof radius e in the Hamming space F n q . The problem of covering codes is an oppositeproblem. Here, we wish to cover all the points of the Hamming space F n q with as fewspheres as possible. Covering codes find applications in data compression.Formally we define:Definition 5Let C be a linear [n, k, d] q -code. The covering radius of the code C is the smallest integerR such that every n-tuple in F n q lies at Hamming distance at most R from a codeword inC.The following theorem will be the basis for making the link with the geometricallyequivalent objects of the saturating sets in finite geometry.Theorem 12Let C be a linear [n, k, d] q -code with parity check matrix H = (h 1 · · · h n ).Then the covering radius of C is equal to R if and only if every (n − k)-tuple overF q can be written as a linear combination of at most R columns of H.Definition 6Let S be a subset of PG(N, q). The set S is called ρ-saturating when every point P fromPG(N, q) can be written as a linear combination of at most ρ + 1 points of S.
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