Best Known (79−39, 79, s)-Nets in Base 32
(79−39, 79, 224)-Net over F32 — Constructive and digital
Digital (40, 79, 224)-net over F32, using
- 1 times m-reduction [i] based on digital (40, 80, 224)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 29, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- digital (11, 51, 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 (9, 29, 104)-net over F32, using
- (u, u+v)-construction [i] based on
(79−39, 79, 288)-Net in Base 32 — Constructive
(40, 79, 288)-net in base 32, using
- 29 times m-reduction [i] based on (40, 108, 288)-net in base 32, using
- base change [i] based on (22, 90, 288)-net in base 64, using
- 1 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- 1 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on (22, 90, 288)-net in base 64, using
(79−39, 79, 687)-Net over F32 — Digital
Digital (40, 79, 687)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3279, 687, F32, 39) (dual of [687, 608, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3279, 1040, F32, 39) (dual of [1040, 961, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(32) [i] based on
- linear OA(3274, 1024, F32, 39) (dual of [1024, 950, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(3263, 1024, F32, 33) (dual of [1024, 961, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(325, 16, F32, 5) (dual of [16, 11, 6]-code or 16-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to Ce(38) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(3279, 1040, F32, 39) (dual of [1040, 961, 40]-code), using
(79−39, 79, 386260)-Net in Base 32 — Upper bound on s
There is no (40, 79, 386261)-net in base 32, because
- 1 times m-reduction [i] would yield (40, 78, 386261)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2521 850084 732726 391206 318392 888410 462100 236099 898354 873033 315629 595525 580084 328943 656171 496776 953528 443350 342459 783708 > 3278 [i]