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
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers O UTLINE 1 C ODING THEORY 2 E QUIVALENT PROBLEM IN CODING THEORY AND G ALOIS GEOMETRIES 3 ( x ( q + 1 ) , x ; 2 , q ) - MINIHYPERS Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers O UTLINE 1 C ODING THEORY 2 E QUIVALENT PROBLEM IN CODING THEORY AND G ALOIS GEOMETRIES 3 ( x ( q + 1 ) , x ; 2 , q ) - MINIHYPERS Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers L INEAR CODES Finite field (Galois field): F q : q prime power, Linear [ n , k , d ] -code C over F q is: k -dimensional subspace of V ( n , q ) , minimal distance d = minimal number of positions in which two distinct codewords differ. T HEOREM 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
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers L INEAR 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
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers R EMARK Remark: For linear [ n , k , d ] -code C , n , k , d do not change when column g i in generator matrix G = ( g 1 · · · g n ) is replaced by non-zero scalar multiple. Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers F ROM VECTOR SPACE TO PROJECTIVE SPACE Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers T HE F ANO PLANE PG ( 2 , 2 ) From V ( 3 , 2 ) to PG ( 2 , 2 ) Leo Storme Coding theory and Galois geometries
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
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers O UTLINE 1 C ODING THEORY 2 E QUIVALENT PROBLEM IN CODING THEORY AND G ALOIS GEOMETRIES 3 ( x ( q + 1 ) , x ; 2 , q ) - MINIHYPERS Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers M INIHYPERS AND G RIESMER BOUND Problem: Given dimension k , minimal distance d , find minimal length n of [ n , k , d ] -code over F q . Result: Griesmer (lower) bound � d k − 1 � � n ≥ = g q ( k , d ) . q i i = 0 Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers M INIHYPERS AND G RIESMER BOUND Equivalence: (Hamada and Helleseth) Griesmer (lower) bound equivalent with minihypers in finite projective spaces Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers D EFINITION D EFINITION { 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
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers M INIHYPERS AND G RIESMER BOUND Let C = [ g q ( k , d ) , k , d ] -code over F q . If generator matrix G = ( g 1 · · · g n ) , minihyper = PG ( k − 1 , q ) \ { g 1 , . . . , g n } . Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers M INIHYPERS AND G RIESMER BOUND Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers E XAMPLES Example 1. Subspace PG ( µ, q ) in PG ( k − 1 , q ) = minihyper of [ n = ( q k − q µ + 1 ) / ( q − 1 ) , k , q k − 1 − q µ ] -code (McDonald code). Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers B OSE -B URTON THEOREM T HEOREM (B OSE -B URTON ) 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
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers T HEOREM For minihyper F in PG ( 2 , q ) intersecting every line in at least one point, | F | ≥ q + 1 and | F | = q + 1 if and only if F is line L. Proof: (1) Let r �∈ F . Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers R AJ C HANDRA B OSE 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. Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers E XAMPLES Example 2. t < q pairwise disjoint subspaces PG ( µ, q ) i , i = 1 , . . . , t , in PG ( k − 1 , q ) = minihyper of [ n = ( q k − 1 ) / ( q − 1 ) − t ( q µ + 1 − 1 ) / ( q − 1 ) , k , q k − 1 − tq µ ] -code. Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers C HARACTERIZATION RESULT T HEOREM (G OVAERTS AND S TORME ) For q odd prime and 1 ≤ t ≤ ( q + 1 ) / 2 , [ n = ( q k − 1 ) / ( q − 1 ) − t ( q µ + 1 − 1 ) / ( q − 1 ) , k , q k − 1 − tq µ ] -code C: minihyper is union of t pairwise disjoint PG ( µ, q ) . T HEOREM (B LOKHUIS ) 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
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers O UTLINE 1 C ODING THEORY 2 E QUIVALENT PROBLEM IN CODING THEORY AND G ALOIS GEOMETRIES 3 ( x ( q + 1 ) , x ; 2 , q ) - MINIHYPERS Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers ( 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 L 1 + · · · + L x . Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers ( x ( q + 1 ) , x ; 2 , q ) - MINIHYPERS Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers ( x ( q + 1 ) , x ; 2 , q ) - MINIHYPERS T HEOREM (H ILL , W ARD ) Let F be an ( x ( q + 1 ) , x ; 2 , q ) -minihyper, q = p m , p prime, m ≥ 1 , with x < q, where p f divides x. Then for each line L in PG ( 2 , q ) , | L ∩ F | ≡ x ( mod p f + 1 ) . T HEOREM (H ILL , W ARD ) Every ( x ( q + 1 ) , x ; 2 , q ) -minihyper F, q even, m ≥ 1 , with x ≤ q / 2 , is a sum of x lines. Leo Storme Coding theory and Galois geometries
Coding theory Equivalent problem in coding theory and Galois geometries ( x ( q + 1 ) , x ; 2 , q ) -minihypers D UAL 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 2 L 1 + · · · + 1 2 L q + 2 . Leo Storme Coding theory and Galois geometries
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