Best Known (34, 34+37, s)-Nets in Base 32
(34, 34+37, 202)-Net over F32 — Constructive and digital
Digital (34, 71, 202)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 25, 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, 46, 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, 25, 98)-net over F32, using
(34, 34+37, 288)-Net in Base 32 — Constructive
(34, 71, 288)-net in base 32, using
- t-expansion [i] based on (33, 71, 288)-net in base 32, using
- 13 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
- 13 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
(34, 34+37, 478)-Net over F32 — Digital
Digital (34, 71, 478)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3271, 478, F32, 2, 37) (dual of [(478, 2), 885, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3271, 515, F32, 2, 37) (dual of [(515, 2), 959, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3271, 1030, F32, 37) (dual of [1030, 959, 38]-code), using
- construction XX applied to C1 = C([1021,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([1021,34]) [i] based on
- linear OA(3268, 1023, F32, 36) (dual of [1023, 955, 37]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,33}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3266, 1023, F32, 35) (dual of [1023, 957, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(3270, 1023, F32, 37) (dual of [1023, 953, 38]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,34}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3264, 1023, F32, 34) (dual of [1023, 959, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [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, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), 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([1021,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([1021,34]) [i] based on
- OOA 2-folding [i] based on linear OA(3271, 1030, F32, 37) (dual of [1030, 959, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(3271, 515, F32, 2, 37) (dual of [(515, 2), 959, 38]-NRT-code), using
(34, 34+37, 173822)-Net in Base 32 — Upper bound on s
There is no (34, 71, 173823)-net in base 32, because
- 1 times m-reduction [i] would yield (34, 70, 173823)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2293 549803 115021 678572 254882 054988 056667 584455 531215 285938 458468 581124 767724 877940 835980 824717 038480 513123 > 3270 [i]