Best Known (78−41, 78, s)-Nets in Base 32
(78−41, 78, 202)-Net over F32 — Constructive and digital
Digital (37, 78, 202)-net over F32, using
- 1 times m-reduction [i] based on digital (37, 79, 202)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 28, 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, 51, 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, 28, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(78−41, 78, 288)-Net in Base 32 — Constructive
(37, 78, 288)-net in base 32, using
- 20 times m-reduction [i] based on (37, 98, 288)-net in base 32, using
- base change [i] based on digital (9, 70, 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, 70, 288)-net over F128, using
(78−41, 78, 477)-Net over F32 — Digital
Digital (37, 78, 477)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3278, 477, F32, 2, 41) (dual of [(477, 2), 876, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3278, 513, F32, 2, 41) (dual of [(513, 2), 948, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3278, 1026, F32, 41) (dual of [1026, 948, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(39) [i] based on
- linear OA(3278, 1024, F32, 41) (dual of [1024, 946, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3276, 1024, F32, 40) (dual of [1024, 948, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- 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 X applied to Ce(40) ⊂ Ce(39) [i] based on
- OOA 2-folding [i] based on linear OA(3278, 1026, F32, 41) (dual of [1026, 948, 42]-code), using
- discarding factors / shortening the dual code based on linear OOA(3278, 513, F32, 2, 41) (dual of [(513, 2), 948, 42]-NRT-code), using
(78−41, 78, 167011)-Net in Base 32 — Upper bound on s
There is no (37, 78, 167012)-net in base 32, because
- 1 times m-reduction [i] would yield (37, 77, 167012)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 78 811646 049388 591127 191532 235361 377720 147826 342380 916580 660367 873852 122310 057421 017542 787943 521767 818738 535947 539784 > 3277 [i]