Best Known (79−36, 79, s)-Nets in Base 32
(79−36, 79, 240)-Net over F32 — Constructive and digital
Digital (43, 79, 240)-net over F32, using
- 6 times m-reduction [i] based on digital (43, 85, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 32, 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, 53, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 32, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(79−36, 79, 513)-Net in Base 32 — Constructive
(43, 79, 513)-net in base 32, using
- 11 times m-reduction [i] based on (43, 90, 513)-net in base 32, using
- base change [i] based on digital (28, 75, 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, 75, 513)-net over F64, using
(79−36, 79, 1169)-Net over F32 — Digital
Digital (43, 79, 1169)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3279, 1169, F32, 36) (dual of [1169, 1090, 37]-code), using
- 131 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0, 1, 60 times 0) [i] based on linear OA(3268, 1027, F32, 36) (dual of [1027, 959, 37]-code), using
- construction XX applied to C1 = C([1022,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([1022,34]) [i] based on
- linear OA(3266, 1023, F32, 35) (dual of [1023, 957, 36]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,33}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(3266, 1023, F32, 35) (dual of [1023, 957, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(3268, 1023, F32, 36) (dual of [1023, 955, 37]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,34}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3264, 1023, F32, 34) (dual of [1023, 959, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [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([1022,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([1022,34]) [i] based on
- 131 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0, 1, 60 times 0) [i] based on linear OA(3268, 1027, F32, 36) (dual of [1027, 959, 37]-code), using
(79−36, 79, 983332)-Net in Base 32 — Upper bound on s
There is no (43, 79, 983333)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 80696 130528 511324 593021 346891 435812 218810 082940 548116 686853 131832 654299 907224 631474 390691 954361 839135 429021 373510 654180 > 3279 [i]