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Coding theory and Galois geometries: two interacting research areas - - PowerPoint PPT Presentation

Oct 11, 2022 450 likes •760 views

Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers Coding theory and Galois geometries: two interacting research areas Leo Storme Ghent University Dept. of Pure Mathematics and

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

Coding theory and Galois geometries: two interacting research areas

Leo Storme

Ghent University

  • Dept. of Pure Mathematics and Computer Algebra

Krijgslaan 281 - S22 9000 Ghent Belgium

XVIII Latin American Algebra Colloquium 2009

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

OUTLINE

1 CODING THEORY 2 EQUIVALENT PROBLEM IN CODING THEORY AND GALOIS

GEOMETRIES

3 (x(q + 1), x; 2, q)-MINIHYPERS

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

OUTLINE

1 CODING THEORY 2 EQUIVALENT PROBLEM IN CODING THEORY AND GALOIS

GEOMETRIES

3 (x(q + 1), x; 2, q)-MINIHYPERS

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

LINEAR CODES

Finite field (Galois field): Fq: q prime power, Linear [n, k, d]-code C over Fq is:

k-dimensional subspace of V(n, q), minimal distance d = minimal number of positions in which two distinct codewords differ.

THEOREM If in transmitted codeword at most (d − 1)/2 errors, it is possible to correct these errors by replacing the received n-tuple by the codeword at minimal distance.

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

LINEAR CODES

Generator matrix G of [n, k, d]-code C

G = (k × n) matrix of rank k, rows of G form a basis of C, codeword of C = linear combination of rows of G.

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

REMARK

Remark: For linear [n, k, d]-code C, n, k, d do not change when column gi in generator matrix G = (g1 · · · gn) is replaced by non-zero scalar multiple.

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

FROM VECTOR SPACE TO PROJECTIVE SPACE

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

THE FANO PLANE PG(2, 2)

From V(3, 2) to PG(2, 2)

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

PG(3, 2)

From V(4, 2) to PG(3, 2)

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

OUTLINE

1 CODING THEORY 2 EQUIVALENT PROBLEM IN CODING THEORY AND GALOIS

GEOMETRIES

3 (x(q + 1), x; 2, q)-MINIHYPERS

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

MINIHYPERS AND GRIESMER BOUND

Problem: Given dimension k, minimal distance d, find minimal length n of [n, k, d]-code over Fq. Result: Griesmer (lower) bound n ≥

k−1

  • i=0

d qi

  • = gq(k, d).

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

MINIHYPERS AND GRIESMER BOUND

Equivalence: (Hamada and Helleseth) Griesmer (lower) bound equivalent with minihypers in finite projective spaces

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

DEFINITION

DEFINITION {f, m; k − 1, q}-minihyper F is: set of f points in PG(k − 1, q), F intersects every (k − 2)-dimensional space in at least m points.

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

MINIHYPERS AND GRIESMER BOUND

Let C = [gq(k, d), k, d]-code over Fq. If generator matrix G = (g1 · · · gn), minihyper = PG(k − 1, q) \ {g1, . . . , gn}.

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

MINIHYPERS AND GRIESMER BOUND

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

EXAMPLES

Example 1. Subspace PG(µ, q) in PG(k − 1, q) = minihyper of [n = (qk − qµ+1)/(q − 1), k, qk−1 − qµ]-code (McDonald code).

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

BOSE-BURTON THEOREM

THEOREM (BOSE-BURTON) A minihyper consisting of |PG(µ, q)| points intersecting every hyperplane in at least |PG(µ − 1, q)| points is equal to a µ-dimensional space PG(µ, q).

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

THEOREM For minihyper F in PG(2, q) intersecting every line in at least

  • ne point, |F| ≥ q + 1 and |F| = q + 1 if and only if F is line L.

Proof: (1) Let r ∈ F.

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

RAJ CHANDRA BOSE

R.C. Bose and R.C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the McDonald codes. J. Combin. Theory, 1:96-104, 1966.

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

EXAMPLES

Example 2. t < q pairwise disjoint subspaces PG(µ, q)i, i = 1, . . . , t, in PG(k − 1, q) = minihyper of [n = (qk − 1)/(q − 1) − t(qµ+1 − 1)/(q − 1), k, qk−1 − tqµ]-code.

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CHARACTERIZATION RESULT

THEOREM (GOVAERTS AND STORME) For q odd prime and 1 ≤ t ≤ (q + 1)/2, [n = (qk − 1)/(q − 1) − t(qµ+1 − 1)/(q − 1), k, qk−1 − tqµ]-code C: minihyper is union of t pairwise disjoint PG(µ, q). THEOREM (BLOKHUIS) A minihyper F in PG(2, q), q odd prime, intersecting every line in at least one point, of size at most q + (q + 1)/2, always contains a line.

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

OUTLINE

1 CODING THEORY 2 EQUIVALENT PROBLEM IN CODING THEORY AND GALOIS

GEOMETRIES

3 (x(q + 1), x; 2, q)-MINIHYPERS

Leo Storme Coding theory and Galois geometries

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(x(q + 1), x; 2, q)-MINIHYPERS

Hill and Ward: detailed study of (x(q + 1), x; 2, q)-minihypers. Weighted set of x(q + 1) points in PG(2, q) intersecting every line in at least x points. Classical example: sum of x (not necessarily distinct) lines L1 + · · · + Lx.

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(x(q + 1), x; 2, q)-MINIHYPERS

Leo Storme Coding theory and Galois geometries

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

(x(q + 1), x; 2, q)-MINIHYPERS

THEOREM (HILL, WARD) Let F be an (x(q + 1), x; 2, q)-minihyper, q = pm, p prime, m ≥ 1, with x < q, where pf divides x. Then for each line L in PG(2, q), |L ∩ F| ≡ x (mod pf+1). THEOREM (HILL, WARD) Every (x(q + 1), x; 2, q)-minihyper F, q even, m ≥ 1, with x ≤ q/2, is a sum of x lines.

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Coding theory Equivalent problem in coding theory and Galois geometries (x(q + 1), x; 2, q)-minihypers

DUAL HYPEROVAL EXAMPLE

Take a dual hyperoval O in PG(2, q), q even: q + 2 lines, no three through a common point. Point of PG(2, q) lies on zero or two lines of O. Union of the lines of dual hyperoval is ((q/2 + 1)(q + 1), q/2 + 1; 2, q)-minihyper. This particular example is not a sum of q/2 + 1 lines with integer coefficients. This particular example is a sum of q + 2 lines with coefficients 1/2: 1 2L1 + · · · + 1 2Lq+2.

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RATIONAL SUM OF LINES

THEOREM (LANDJEV, STORME) Let F be an (x(q + 1), x; 2, q)-minihyper. Then there exist lines L1, . . . , Ls and positive rational numbers c1, . . . , cs, such that F = c1L1 + · · · + csLs, with s

i=1 ci = x.

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CHARACTERIZATION RESULT

THEOREM (LANDJEV, STORME, METSCH) Every ((q/2 + 1)(q + 1), q/2 + 1; 2, q)-minihyper, q even, is either: sum of q/2 + 1 lines, a sum 1 2L1 + · · · + 1 2Lq+2, where {L1, . . . , Lq+2} is dual hyperoval.

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Thank you very much for your attention!

Leo Storme Coding theory and Galois geometries