Best Known (76−39, 76, s)-Nets in Base 32
(76−39, 76, 218)-Net over F32 — Constructive and digital
Digital (37, 76, 218)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 26, 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, 50, 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, 26, 98)-net over F32, using
(76−39, 76, 288)-Net in Base 32 — Constructive
(37, 76, 288)-net in base 32, using
- 22 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
(76−39, 76, 516)-Net over F32 — Digital
Digital (37, 76, 516)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3276, 516, F32, 2, 39) (dual of [(516, 2), 956, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3276, 1032, F32, 39) (dual of [1032, 956, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(35) [i] based on
- linear OA(3274, 1024, F32, 39) (dual of [1024, 950, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(3268, 1024, F32, 36) (dual of [1024, 956, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(322, 8, F32, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(38) ⊂ Ce(35) [i] based on
- OOA 2-folding [i] based on linear OA(3276, 1032, F32, 39) (dual of [1032, 956, 40]-code), using
(76−39, 76, 223468)-Net in Base 32 — Upper bound on s
There is no (37, 76, 223469)-net in base 32, because
- 1 times m-reduction [i] would yield (37, 75, 223469)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 76959 268858 637812 376582 161156 779527 625775 029475 298597 630415 887293 619022 603298 525940 957949 318113 137114 191272 860536 > 3275 [i]