Best Known (63−31, 63, s)-Nets in Base 32
(63−31, 63, 202)-Net over F32 — Constructive and digital
Digital (32, 63, 202)-net over F32, using
- 1 times m-reduction [i] based on digital (32, 64, 202)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 23, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (9, 41, 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 (7, 23, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(63−31, 63, 288)-Net in Base 32 — Constructive
(32, 63, 288)-net in base 32, using
- t-expansion [i] based on (31, 63, 288)-net in base 32, using
- 14 times m-reduction [i] based on (31, 77, 288)-net in base 32, using
- base change [i] based on digital (9, 55, 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, 55, 288)-net over F128, using
- 14 times m-reduction [i] based on (31, 77, 288)-net in base 32, using
(63−31, 63, 609)-Net over F32 — Digital
Digital (32, 63, 609)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3263, 609, F32, 31) (dual of [609, 546, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3263, 1032, F32, 31) (dual of [1032, 969, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3261, 1024, F32, 31) (dual of [1024, 963, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3255, 1024, F32, 28) (dual of [1024, 969, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(322, 8, F32, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3263, 1032, F32, 31) (dual of [1032, 969, 32]-code), using
(63−31, 63, 344891)-Net in Base 32 — Upper bound on s
There is no (32, 63, 344892)-net in base 32, because
- 1 times m-reduction [i] would yield (32, 62, 344892)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2085 998359 886945 634067 050598 033485 419600 234807 543777 940207 579850 102525 866349 767131 981026 789108 > 3262 [i]