Best Known (66−32, 66, s)-Nets in Base 32
(66−32, 66, 218)-Net over F32 — Constructive and digital
Digital (34, 66, 218)-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 (11, 43, 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 (7, 23, 98)-net over F32, using
(66−32, 66, 288)-Net in Base 32 — Constructive
(34, 66, 288)-net in base 32, using
- t-expansion [i] based on (33, 66, 288)-net in base 32, using
- 18 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
- base change [i] based on digital (9, 60, 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, 60, 288)-net over F128, using
- 18 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
(66−32, 66, 695)-Net over F32 — Digital
Digital (34, 66, 695)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3266, 695, F32, 32) (dual of [695, 629, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3266, 1036, F32, 32) (dual of [1036, 970, 33]-code), using
- construction XX applied to C1 = C([1019,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([1019,27]) [i] based on
- linear OA(3261, 1023, F32, 31) (dual of [1023, 962, 32]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−4,−3,…,26}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3255, 1023, F32, 28) (dual of [1023, 968, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3263, 1023, F32, 32) (dual of [1023, 960, 33]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−4,−3,…,27}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3253, 1023, F32, 27) (dual of [1023, 970, 28]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(323, 11, F32, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- 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
- construction XX applied to C1 = C([1019,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([1019,27]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3266, 1036, F32, 32) (dual of [1036, 970, 33]-code), using
(66−32, 66, 354740)-Net in Base 32 — Upper bound on s
There is no (34, 66, 354741)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2187 291680 574261 869554 670527 848191 002428 366512 807730 115825 949050 857495 900904 350272 600941 813308 293451 > 3266 [i]