Best Known (35, 35+31, s)-Nets in Base 32
(35, 35+31, 224)-Net over F32 — Constructive and digital
Digital (35, 66, 224)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 24, 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, 42, 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, 24, 104)-net over F32, using
(35, 35+31, 288)-Net in Base 32 — Constructive
(35, 66, 288)-net in base 32, using
- 25 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
- base change [i] based on digital (9, 65, 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, 65, 288)-net over F128, using
(35, 35+31, 876)-Net over F32 — Digital
Digital (35, 66, 876)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3266, 876, F32, 31) (dual of [876, 810, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3266, 1042, F32, 31) (dual of [1042, 976, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3261, 1025, F32, 31) (dual of [1025, 964, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3249, 1025, F32, 25) (dual of [1025, 976, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(325, 17, F32, 5) (dual of [17, 12, 6]-code or 17-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 C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3266, 1042, F32, 31) (dual of [1042, 976, 32]-code), using
(35, 35+31, 689789)-Net in Base 32 — Upper bound on s
There is no (35, 66, 689790)-net in base 32, because
- 1 times m-reduction [i] would yield (35, 65, 689790)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 68 351788 762757 623106 099381 454875 264174 791746 627700 323250 601662 650176 421346 462773 682116 640249 430728 > 3265 [i]