Best Known (80−42, 80, s)-Nets in Base 32
(80−42, 80, 202)-Net over F32 — Constructive and digital
Digital (38, 80, 202)-net over F32, using
- 2 times m-reduction [i] based on digital (38, 82, 202)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 29, 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, 53, 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, 29, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(80−42, 80, 288)-Net in Base 32 — Constructive
(38, 80, 288)-net in base 32, using
- t-expansion [i] based on (37, 80, 288)-net in base 32, using
- 18 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
- 18 times m-reduction [i] based on (37, 98, 288)-net in base 32, using
(80−42, 80, 489)-Net over F32 — Digital
Digital (38, 80, 489)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3280, 489, F32, 2, 42) (dual of [(489, 2), 898, 43]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3280, 513, F32, 2, 42) (dual of [(513, 2), 946, 43]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3280, 1026, F32, 42) (dual of [1026, 946, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(40) [i] based on
- linear OA(3280, 1024, F32, 42) (dual of [1024, 944, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- 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(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(41) ⊂ Ce(40) [i] based on
- OOA 2-folding [i] based on linear OA(3280, 1026, F32, 42) (dual of [1026, 946, 43]-code), using
- discarding factors / shortening the dual code based on linear OOA(3280, 513, F32, 2, 42) (dual of [(513, 2), 946, 43]-NRT-code), using
(80−42, 80, 513)-Net in Base 32
(38, 80, 513)-net in base 32, using
- base change [i] based on digital (8, 50, 513)-net over F256, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- K1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F256 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
(80−42, 80, 151706)-Net in Base 32 — Upper bound on s
There is no (38, 80, 151707)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2 582328 852200 222753 814690 502394 282589 200953 125655 973363 561489 504554 988251 372482 745193 571418 588094 356836 148474 503417 137324 > 3280 [i]