Best Known (77−37, 77, s)-Nets in Base 32
(77−37, 77, 240)-Net over F32 — Constructive and digital
Digital (40, 77, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 29, 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, 48, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 29, 120)-net over F32, using
(77−37, 77, 288)-Net in Base 32 — Constructive
(40, 77, 288)-net in base 32, using
- 31 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
(77−37, 77, 816)-Net over F32 — Digital
Digital (40, 77, 816)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3277, 816, F32, 37) (dual of [816, 739, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3277, 1044, F32, 37) (dual of [1044, 967, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- linear OA(3270, 1024, F32, 37) (dual of [1024, 954, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3257, 1024, F32, 29) (dual of [1024, 967, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(327, 20, F32, 7) (dual of [20, 13, 8]-code or 20-arc in PG(6,32)), using
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- Reed–Solomon code RS(25,32) [i]
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3277, 1044, F32, 37) (dual of [1044, 967, 38]-code), using
(77−37, 77, 551872)-Net in Base 32 — Upper bound on s
There is no (40, 77, 551873)-net in base 32, because
- 1 times m-reduction [i] would yield (40, 76, 551873)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2 462650 275600 898778 163717 227121 534626 670866 865478 762181 939788 337958 091447 692078 020970 621240 945153 616129 327972 459168 > 3276 [i]