Best Known (56, 56+51, s)-Nets in Base 32
(56, 56+51, 240)-Net over F32 — Constructive and digital
Digital (56, 107, 240)-net over F32, using
- t-expansion [i] based on digital (51, 107, 240)-net over F32, using
- 2 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 40, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (11, 69, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 40, 120)-net over F32, using
- (u, u+v)-construction [i] based on
- 2 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(56, 56+51, 513)-Net in Base 32 — Constructive
(56, 107, 513)-net in base 32, using
- t-expansion [i] based on (46, 107, 513)-net in base 32, using
- 1 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- 1 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(56, 56+51, 1089)-Net over F32 — Digital
Digital (56, 107, 1089)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32107, 1089, F32, 51) (dual of [1089, 982, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(32107, 1101, F32, 51) (dual of [1101, 994, 52]-code), using
- 65 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 16 times 0, 1, 32 times 0) [i] based on linear OA(3298, 1027, F32, 51) (dual of [1027, 929, 52]-code), using
- construction XX applied to C1 = C([1022,48]), C2 = C([0,49]), C3 = C1 + C2 = C([0,48]), and C∩ = C1 ∩ C2 = C([1022,49]) [i] based on
- linear OA(3296, 1023, F32, 50) (dual of [1023, 927, 51]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,48}, and designed minimum distance d ≥ |I|+1 = 51 [i]
- linear OA(3296, 1023, F32, 50) (dual of [1023, 927, 51]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,49], and designed minimum distance d ≥ |I|+1 = 51 [i]
- linear OA(3298, 1023, F32, 51) (dual of [1023, 925, 52]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,49}, and designed minimum distance d ≥ |I|+1 = 52 [i]
- linear OA(3294, 1023, F32, 49) (dual of [1023, 929, 50]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,48], and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,48]), C2 = C([0,49]), C3 = C1 + C2 = C([0,48]), and C∩ = C1 ∩ C2 = C([1022,49]) [i] based on
- 65 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 16 times 0, 1, 32 times 0) [i] based on linear OA(3298, 1027, F32, 51) (dual of [1027, 929, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(32107, 1101, F32, 51) (dual of [1101, 994, 52]-code), using
(56, 56+51, 790847)-Net in Base 32 — Upper bound on s
There is no (56, 107, 790848)-net in base 32, because
- 1 times m-reduction [i] would yield (56, 106, 790848)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 3514 778799 622978 029406 374847 123746 340076 649525 856528 111735 845676 430373 329937 669085 568566 616512 055285 640305 407980 366588 980886 493840 684399 763306 849064 504747 500285 > 32106 [i]