Galois geometries contributing to coding theory
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1 Ghent University Dept. of Mathematics Krijgslaan S Ghent Belgium Opatija, 2010
2 OUTLINE 1 CODING THEORY 2 GRIESMER BOUND AND MINIHYPERS 3 COVERING RADIUS AND SATURATING SETS 4 LINEAR MDS CODES AND ARCS 5 EXTENDABILITY RESULTS AND BLOCKING SETS
3 OUTLINE 1 CODING THEORY 2 GRIESMER BOUND AND MINIHYPERS 3 COVERING RADIUS AND SATURATING SETS 4 LINEAR MDS CODES AND ARCS 5 EXTENDABILITY RESULTS AND BLOCKING SETS
4 LINEAR CODES q = prime number, Prime fields: F q = {1,..., q} (mod q), Finite fields (Galois fields): F q : q prime power, Linear [n, k, d]-code C over F q is: k-dimensional subspace of V (n, q), minimum distance d = minimal number of positions in which two distinct codewords differ.
5 EXAMPLE Example [5, 1, 5]-code over F 2 ; yes = (0, 0, 0, 0, 0), no = (1, 1, 1, 1, 1). (0, 0, 0, 0, 1) or (0, 0, 0, 1, 1) received, most likely (0, 0, 0, 0, 0) = yes transmitted. 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.
6 LINEAR CODES Generator matrix of [n, k, d]-code C G = (g 1 g n ) G = (k n) matrix of rank k, rows of G form basis of C, codeword of C = linear combination of rows of G.
7 EXAMPLE OF GENERATOR MATRIX Matrix G = generates [7, 4, 3]-code over F
8 LINEAR CODES Parity check matrix H for C (n k) n matrix of rank n k, c C c H T = 0.
9 EXAMPLE OF PARITY CHECK MATRIX Matrix H = is parity check matrix for [7, 4, 3]-code over F 2.
10 REMARK 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.
11 FROM VECTOR SPACE TO PROJECTIVE SPACE
12
13 THE FANO PLANE PG(2, 2) From V (3, 2) to PG(2, 2)
14 PG(3, 2) Coding theory From V (4, 2) to PG(3, 2)
15 OUTLINE 1 CODING THEORY 2 GRIESMER BOUND AND MINIHYPERS 3 COVERING RADIUS AND SATURATING SETS 4 LINEAR MDS CODES AND ARCS 5 EXTENDABILITY RESULTS AND BLOCKING SETS
16 GRIESMER BOUND AND MINIHYPERS Question: Given dimension k, minimal distance d, find minimal length n of [n, k, d]-code over F q. Result: Griesmer (lower) bound k 1 d n q i = g q (k, d). i=0
17 MINIHYPERS AND GRIESMER BOUND Equivalence: (Hamada and Helleseth) Griesmer (lower) bound equivalent with minihypers in finite projective spaces
18 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. (m-fold blocking sets with respect to the hyperplanes of PG(k 1, q))
19 MINIHYPERS AND GRIESMER 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 }.
20 MINIHYPERS AND GRIESMER BOUND
21 EXAMPLE Example: Griesmer [8,4,4]-code over F 2 G = minihyper = PG(3, 2)\ {columns of G} = plane (X 0 = 0).
22 CORRESPONDING MINIHYPER
23 OTHER EXAMPLES 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).
24 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).
25 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.
26 OTHER EXAMPLES 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.
27 CHARACTERIZATION RESULT THEOREM (GOVAERTS AND STORME) 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).
28 OUTLINE 1 CODING THEORY 2 GRIESMER BOUND AND MINIHYPERS 3 COVERING RADIUS AND SATURATING SETS 4 LINEAR MDS CODES AND ARCS 5 EXTENDABILITY RESULTS AND BLOCKING SETS
29 DEFINITION DEFINITION Let C be linear [n, k, d]-code over F q. The covering radius of C is smallest integer R such that every n-tuple in F n q differs in at most R positions from some codeword in C. THEOREM Let C be linear [n, k, d]-code over F q with parity check matrix H = (h 1 h n ). Then covering radius of C is equal to R if and only if every (n k)-tuple over F q can be written as linear combination of at most R columns of H.
30 DEFINITION DEFINITION Let S be subset of PG(N, q). The set S is called ρ-saturating when every point P from PG(N, q) can be written as linear combination of at most ρ + 1 points of S. Covering radius ρ for linear [n, k, d]-code equivalent with (ρ 1)-saturating set in PG(n k 1, q)
31 1-SATURATING SETS H = (h 1 h n )
32 2-SATURATING SETS H = (h 1 h n )
33 1-SATURATING SET IN PG(3, q) OF SIZE 2q + 2
34 1-SATURATING SET IN PG(3, q) OF SIZE 2q + 2
35 EXAMPLE OF ÖSTERGÅRD AND DAVYDOV Let F q = {a 1 = 0, a 2,..., a q }. H 1 = a 1 a q a1 2 aq a 2 a q Columns of H 1 define 1-saturating set of size 2q + 1 in PG(3, q).
36 EXAMPLE OF ÖSTERGÅRD AND DAVYDOV
37 EXAMPLE OF ÖSTERGÅRD AND DAVYDOV H 2 = a 1 a q a1 2 aq a 2 a q a1 2 aq a 1 a q Columns of H 2 define 2-saturating set of size 3q + 1 in PG(5, q).,
38 EXAMPLE OF ÖSTERGÅRD AND DAVYDOV
39 OUTLINE 1 CODING THEORY 2 GRIESMER BOUND AND MINIHYPERS 3 COVERING RADIUS AND SATURATING SETS 4 LINEAR MDS CODES AND ARCS 5 EXTENDABILITY RESULTS AND BLOCKING SETS
40 LINEAR MDS CODES AND ARCS Question: Given length n, dimension k, find maximal value of d. Result: Singleton (upper) bound d n k + 1. Notation: MDS code = [n, k, n k + 1]-code.
41 ARCS Equivalence: Singleton (upper) bound (MDS codes) equivalent with Arcs in finite projective spaces (Segre)
42 DEFINITION DEFINITION n-arc in PG(k 1, q): set of n points, every k linearly independent. Example: 1 n-arc in PG(2, q): n points, no three collinear. 2 Conic X 2 1 = X 0X 2 {(1, t, t 2 ) t F q } {(0, 0, 1)}
43 NORMAL RATIONAL CURVE Classical example of arc: {(1, t,..., t k 1 ) t F q } {(0,..., 0, 1)} defines [q + 1, k, d = q + 2 k]-gdrs (Generalized Doubly-Extended Reed-Solomon) code with generator matrix G = t 1 t q 0... t k 2 1 t k 2 t k 1 1 t k 1. q 0 q 1
44 CHARACTERIZATION RESULT THEOREM (SEGRE, THAS) For q odd prime power, 2 k < q/4, [n = q + 1, k, d = q + 2 k]-mds code is GDRS.
45 BALL RESULT THEOREM (BALL) For q odd prime, n q + 1 for every [n, k, n k + 1]-MDS code. Technique: Polynomial techniques
46 OUTLINE 1 CODING THEORY 2 GRIESMER BOUND AND MINIHYPERS 3 COVERING RADIUS AND SATURATING SETS 4 LINEAR MDS CODES AND ARCS 5 EXTENDABILITY RESULTS AND BLOCKING SETS
47 WELL-KNOWN EXTENDABILITY RESULT THEOREM Every linear binary [n, k, d]-code C, d odd, is extendable to linear binary [n + 1, k, d + 1]-code.
48 HILL-LIZAK RESULT THEOREM (HILL AND LIZAK) Let C be linear [n, k, d]-code over F q, with gcd(d, q) = 1 and with all weights congruent to 0 or d (mod q). Then C can be extended to [n + 1, k, d + 1]-code Ĉ all of whose weights are congruent to 0 or d + 1 (mod q). Proof: Subcode of all codewords of weight congruent to 0 (mod q) is linear subcode C 0 of dimension k 1. If G 0 defines C 0 and ( ) x G =, then G 0
49 HILL-LIZAK RESULT Ĝ = x 1 0 G 0. 0 defines Ĉ.
50 GEOMETRICAL COUNTERPART OF LANDJEV DEFINITION Multiset K in PG(k 1, q) is (n, w; k 1, q)-multiarc or (n, w; k 1, q)-arc if 1 sum of all weights of points of K is n, 2 hyperplane H of PG(k 1, q) contains at most w (weighted) points of K and some hyperplane H 0 contains w (weighted) points of K.
51 LINEAR CODES AND MULTIARCS Let C = [n, k, d]-code over F q. If generator matrix G = (g 1 g n ), then {g 1,..., g n } = (n, w = n d; k 1, q)-multiarc.
52 LINEAR CODES AND MULTIARCS
53 GEOMETRICAL COUNTERPART OF LANDJEV C linear [n, k, d]-code over F q, gcd(d, q) = 1 and with all weights congruent to 0 or d (mod q). Then C can be extended to [n + 1, k, d + 1]-code all of whose weights are congruent to 0 or d + 1 (mod q). K =(n, w; k 1, q)-multiarc with gcd(n w, q) = 1 and intersection size of K with all hyperplanes congruent to n or w (mod q). Then K can be extended to (n + 1, w; k 1, q)-multiarc.
54 GEOMETRICAL COUNTERPART OF LANDJEV Proof: Hyperplanes H containing n (mod q) points of K form dual blocking set B with respect to codimension 2 subspaces of PG(k 1, q). Also B = qk 1 1 q 1. By dual of Bose-Burton, B consists of all hyperplanes through particular point P. This point P extends K to (n + 1, w; k 1, q)-multiarc.
55 BLOCKING SETS IN PG(2, q) DEFINITION Blocking set B in PG(2, q): intersects every line in at least one point. Trivial example: Line. DEFINITION Non-trivial blocking set in PG(2, q): contains no line.
56 BLOCKING SETS IN PG(2, q) q + r(q) + 1 = size of smallest non-trivial blocking set in PG(2, q). (Blokhuis) r(q) = (q + 1)/2 for q > 2 prime, (Bruen) r(q) = q + 1 for q square, (Blokhuis) r(q) = q 2/3 + 1 for q cube (non-square) power.
57 IMPROVED RESULTS THEOREM (LANDJEV AND ROUSSEVA) Let K be (n, w; k 1, q)-arc, q = p s, with spectrum (a i ) i 0. Let w n (mod q) and i w (mod q) a i < q k 2 + q k q k 3 r(q), (1) where q + r(q) + 1 is minimal size of non-trivial blocking set of PG(2, q). Then K is extendable to (n + 1, w; k 1, q)-arc.
58 IMPROVED RESULTS THEOREM Let C be non-extendable [n, k, d]-code over F q, q = p s, with gcd(d, q) = 1. If (A i ) i 0 is spectrum of C, then i 0,d (mod q) A i q k 3 r(q), where q + r(q) + 1 is minimal size of non-trivial blocking set of PG(2, q).
59 Thank you very much for your attention!
Galois geometries contributing to coding theory
Ghent University Dept. of Pure Mathematics and Computer Algebra Krijgslaan 281 - S22 9000 Ghent Belgium Thurnau, April 15, 2010 OUTLINE 1 CODING THEORY 2 GRIESMER BOUND AND MINIHYPERS 3 EXTENDABILITY RESULTS
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