Best Known (56, 109, s)-Nets in Base 32
(56, 109, 240)-Net over F32 — Constructive and digital
Digital (56, 109, 240)-net over F32, using
- t-expansion [i] based on digital (51, 109, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 40, 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 (11, 69, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 40, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(56, 109, 513)-Net in Base 32 — Constructive
(56, 109, 513)-net in base 32, using
- 321 times duplication [i] based on (55, 108, 513)-net in base 32, using
- t-expansion [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- t-expansion [i] based on (46, 108, 513)-net in base 32, using
(56, 109, 962)-Net over F32 — Digital
Digital (56, 109, 962)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32109, 962, F32, 53) (dual of [962, 853, 54]-code), using
- discarding factors / shortening the dual code based on linear OA(32109, 1047, F32, 53) (dual of [1047, 938, 54]-code), using
- construction X applied to Ce(52) ⊂ Ce(44) [i] based on
- linear OA(32102, 1024, F32, 53) (dual of [1024, 922, 54]-code), using an extension Ce(52) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,52], and designed minimum distance d ≥ |I|+1 = 53 [i]
- linear OA(3286, 1024, F32, 45) (dual of [1024, 938, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(327, 23, F32, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,32)), using
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- Reed–Solomon code RS(25,32) [i]
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- construction X applied to Ce(52) ⊂ Ce(44) [i] based on
- discarding factors / shortening the dual code based on linear OA(32109, 1047, F32, 53) (dual of [1047, 938, 54]-code), using
(56, 109, 608251)-Net in Base 32 — Upper bound on s
There is no (56, 109, 608252)-net in base 32, because
- 1 times m-reduction [i] would yield (56, 108, 608252)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 3 599190 353737 908233 828342 698202 770358 978829 185566 578186 566203 851041 645192 195175 828547 138142 319788 147566 738434 923311 906564 594469 595273 162633 277213 928993 981651 943181 > 32108 [i]