Heuristic Construction of Linear Codes over Finite Chain Rings with High Minimum Homogeneous Weight
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1 Heuristic Construction of Linear Codes over Finite Chain Rings with High Minimum Homogeneous Weight Johannes Zwanzger University of Bayreuth Magdeburg November 13th, 2009
2 Motivation J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
3 Motivation 1967: Nordstrom and Robinson find nonlinear binary [16, 2 8, 6]-code (linear: minimum distance 5) J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
4 Motivation 1967: Nordstrom and Robinson find nonlinear binary [16, 2 8, 6]-code (linear: minimum distance 5) Preparata & Kerdock construct infinite series of such codes including the NR-code. J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
5 Motivation 1967: Nordstrom and Robinson find nonlinear binary [16, 2 8, 6]-code (linear: minimum distance 5) Preparata & Kerdock construct infinite series of such codes including the NR-code. Nechaev 1989, Hammons et al. 1994: K. & P.-codes are the Gray image of Z 4 -linear codes J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
6 Motivation 1967: Nordstrom and Robinson find nonlinear binary [16, 2 8, 6]-code (linear: minimum distance 5) Preparata & Kerdock construct infinite series of such codes including the NR-code. Nechaev 1989, Hammons et al. 1994: K. & P.-codes are the Gray image of Z 4 -linear codes Questions: Are there more examples? What about other rings? J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
7 Finite chain rings J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
8 Finite chain rings Definition A finite chain ring R is a finite ring with unity whose left ideals form a chain R = I 0 I 1 I m = {0}. m is called the chain length of R. J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
9 Finite chain rings Definition A finite chain ring R is a finite ring with unity whose left ideals form a chain R = I 0 I 1 I m = {0}. m is called the chain length of R. q : R/I 1 = Fq. J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
10 Finite chain rings Definition A finite chain ring R is a finite ring with unity whose left ideals form a chain R = I 0 I 1 I m = {0}. m is called the chain length of R. q : R/I 1 = Fq. R, m and q will keep their meaning throughout this talk. J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
11 Finite chain rings Definition A finite chain ring R is a finite ring with unity whose left ideals form a chain R = I 0 I 1 I m = {0}. m is called the chain length of R. q : R/I 1 = Fq. R, m and q will keep their meaning throughout this talk. Example R := Z p n, with p a prime (m = n, q = p). J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
12 Linear codes over finite chain rings J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
13 Linear codes over finite chain rings Linear code over R of length n: submodule of R R n (row convention). J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
14 Linear codes over finite chain rings Linear code over R of length n: submodule of R R n (row convention). Example J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
15 Linear codes over finite chain rings Linear code over R of length n: submodule of R R n (row convention). Example R := Z 8 J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
16 Linear codes over finite chain rings Linear code over R of length n: submodule of R R n (row convention). Example R := Z 8 I 0 = Z 8, I 1 = {0, 2, 4, 6}, I 2 = {0, 4}, I 3 = {0} J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
17 Linear codes over finite chain rings Linear code over R of length n: submodule of R R n (row convention). Example R := Z 8 I 0 = Z 8, I 1 = {0, 2, 4, 6}, I 2 = {0, 4}, I 3 = {0} Γ = J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
18 Linear codes over finite chain rings Linear code over R of length n: submodule of R R n (row convention). Example R := Z 8 I 0 = Z 8, I 1 = {0, 2, 4, 6}, I 2 = {0, 4}, I 3 = {0} Γ = C Γ has shape (3, 2, 1). J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
19 Linear codes over finite chain rings Linear code over R of length n: submodule of R R n (row convention). Example R := Z 8 I 0 = Z 8, I 1 = {0, 2, 4, 6}, I 2 = {0, 4}, I 3 = {0} Γ = C Γ has shape (3, 2, 1). R 0 = Z 8, R 1 = {0, 1, 2, 3}, R 2 = {0, 1} J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
20 Linear codes over finite chain rings Linear code over R of length n: submodule of R R n (row convention). Example R := Z 8 I 0 = Z 8, I 1 = {0, 2, 4, 6}, I 2 = {0, 4}, I 3 = {0} Γ = C Γ has shape (3, 2, 1). R 0 = Z 8, R 1 = {0, 1, 2, 3}, R 2 = {0, 1} C Γ = {uγ : u R 0 R 1 R 2 } J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
21 The homogeneous weight J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
22 The homogeneous weight Definition J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
23 The homogeneous weight Definition Homogeneous weight: 0 r = 0 w : R Q, w(r) = q r I m 1 \ {0} q 1 r R \ I m 1 J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
24 The homogeneous weight Definition Homogeneous weight: 0 r = 0 w : R Q, w(r) = q r I m 1 \ {0} q 1 r R \ I m 1 d(r, r ) := w(r r ) (extend to R n ) J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
25 The homogeneous weight Definition Homogeneous weight: 0 r = 0 w : R Q, w(r) = q r I m 1 \ {0} q 1 r R \ I m 1 d(r, r ) := w(r r ) (extend to R n ) Minimum homogeneous distance: d(c) := min{d(c, c ) : c, c C, c c } = min{w(c) : c C \ {0}} J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
26 The homogeneous weight Definition Homogeneous weight: 0 r = 0 w : R Q, w(r) = q r I m 1 \ {0} q 1 r R \ I m 1 d(r, r ) := w(r r ) (extend to R n ) Minimum homogeneous distance: d(c) := min{d(c, c ) : c, c C, c c } = min{w(c) : c C \ {0}} M. Greferath and S. Schmidt, 1999: isometry Ψ : (R, q m 2 d) (F qm 1 q, d ham ) ( Gray map ) J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
27 System of Diophantine inequalities J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
28 System of Diophantine inequalities Theorem Searching a linear (n, λ, δ, R)-code Solving the system: δ Mx. δ 1 T x = n (M N t t 0, x N t 0 ) J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
29 Example J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
30 Example R := Z 4, λ := (2, 1, 1) J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
31 Heuristic algorithm J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
32 Heuristic algorithm Define heuristic evaluation function eval : N t 0 R. J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
33 Heuristic algorithm Define heuristic evaluation function eval : N t 0 R. Construct x 0 x 1 x 2 x n in N t 0 with J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
34 Heuristic algorithm Define heuristic evaluation function eval : N t 0 R. Construct x 0 x 1 x 2 x n in N t 0 with x 0 = 0 J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
35 Heuristic algorithm Define heuristic evaluation function eval : N t 0 R. Construct x 0 x 1 x 2 x n in N t 0 with x 0 = 0 x i+1 = x i + e si J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
36 Heuristic algorithm Define heuristic evaluation function eval : N t 0 R. Construct x 0 x 1 x 2 x n in N t 0 with x 0 = 0 x i+1 = x i + e si s i chosen s. t. eval(x i + e si ) is maximized ( greedy ) J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
37 Heuristic algorithm Define heuristic evaluation function eval : N t 0 R. Construct x 0 x 1 x 2 x n in N t 0 with x 0 = 0 x i+1 = x i + e si s i chosen s. t. eval(x i + e si ) is maximized ( greedy ) If x n is a solution terminate; otherwise: backtracking. J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
38 Evaluation function J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
39 Evaluation function Idea: eval(x)! ɛ(x) := {y x : y 1 = n, My δ 1} {y x : y 1 = n} J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
40 Evaluation function Idea: eval(x)! ɛ(x) := {y x : y 1 = n, My δ 1} {y x : y 1 = n} Consider inequalities M i, y δ separately: J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
41 Evaluation function Idea: eval(x)! ɛ(x) := {y x : y 1 = n, My δ 1} {y x : y 1 = n} Consider inequalities M i, y δ separately: ɛ i (x) := {y x : y 1 = n, M i, y δ} {y x : x 1 = n} J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
42 Evaluation function Idea: eval(x)! ɛ(x) := {y x : y 1 = n, My δ 1} {y x : y 1 = n} Consider inequalities M i, y δ separately: ɛ i (x) := {y x : y 1 = n, M i, y δ} {y x : x 1 = n} Assuming stochastic independence t 1 ɛ(x) ɛ i (x) =: eval(x) i=0 J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
43 Computation of ɛ i J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
44 Computation of ɛ i ɛ i (x) only depends from x 1 and M i, x. J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
45 Computation of ɛ i ɛ i (x) only depends from x 1 and M i, x. Derive lookup table from coefficients of p i (y, z) = n ( ) j + a y sj z j s i 1 j s:a s i >0 j=0 (a s i := multiplicity of weight s in M i, ) J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
46 Computation of ɛ i ɛ i (x) only depends from x 1 and M i, x. Derive lookup table from coefficients of p i (y, z) = n ( ) j + a y sj z j s i 1 j s:a s i >0 j=0 (a s i := multiplicity of weight s in M i, ) Using reductions: (δ + 2)(n + 1) 2 multiplications. J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
47 Results J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
48 Results Codes of high minimum distance for more than 300 pairs (R, λ) (n 100) were constructed. J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
49 Results Codes of high minimum distance for more than 300 pairs (R, λ) (n 100) were constructed. For char(r) = p many of them are provably optimal. J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
50 Results Codes of high minimum distance for more than 300 pairs (R, λ) (n 100) were constructed. For char(r) = p many of them are provably optimal. Two new Z 4 -linear codes whose Gray image beats the corresponding upper binary linear bound were found: J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
51 Results Codes of high minimum distance for more than 300 pairs (R, λ) (n 100) were constructed. For char(r) = p many of them are provably optimal. Two new Z 4 -linear codes whose Gray image beats the corresponding upper binary linear bound were found: λ = (2, 2, 2, 1), n = 29, d(c) = 28 λ = (2, 2, 2, 2), n = 57, d(c) = 56 J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
52 Results Codes of high minimum distance for more than 300 pairs (R, λ) (n 100) were constructed. For char(r) = p many of them are provably optimal. Two new Z 4 -linear codes whose Gray image beats the corresponding upper binary linear bound were found: λ = (2, 2, 2, 1), n = 29, d(c) = 28 λ = (2, 2, 2, 2), n = 57, d(c) = 56 All results: J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
53 Results Codes of high minimum distance for more than 300 pairs (R, λ) (n 100) were constructed. For char(r) = p many of them are provably optimal. Two new Z 4 -linear codes whose Gray image beats the corresponding upper binary linear bound were found: λ = (2, 2, 2, 1), n = 29, d(c) = 28 λ = (2, 2, 2, 2), n = 57, d(c) = 56 All results: Thanks for your attention! J. Zwanzger (Univ. of Bayreuth) Linear Codes over Finite Chain Rings Magdeburg, / 11
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