Best Known (86−43, 86, s)-Nets in Base 32
(86−43, 86, 240)-Net over F32 — Constructive and digital
Digital (43, 86, 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, 54, 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
(86−43, 86, 513)-Net in Base 32 — Constructive
(43, 86, 513)-net in base 32, using
- 4 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
(86−43, 86, 669)-Net over F32 — Digital
Digital (43, 86, 669)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3286, 669, F32, 43) (dual of [669, 583, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3286, 1030, F32, 43) (dual of [1030, 944, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,20]) [i] based on
- linear OA(3285, 1025, F32, 43) (dual of [1025, 940, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(3281, 1025, F32, 41) (dual of [1025, 944, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,21]) ⊂ C([0,20]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3286, 1030, F32, 43) (dual of [1030, 944, 44]-code), using
(86−43, 86, 346251)-Net in Base 32 — Upper bound on s
There is no (43, 86, 346252)-net in base 32, because
- 1 times m-reduction [i] would yield (43, 85, 346252)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 86 650484 647066 069421 397684 957516 530886 024130 287933 669977 060311 416047 903941 562882 596395 546280 107569 672626 676459 024217 320937 389808 > 3285 [i]