Best Known (51−25, 51, s)-Nets in Base 32
(51−25, 51, 196)-Net over F32 — Constructive and digital
Digital (26, 51, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 19, 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 (7, 32, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 19, 98)-net over F32, using
(51−25, 51, 288)-Net in Base 32 — Constructive
(26, 51, 288)-net in base 32, using
- t-expansion [i] based on (25, 51, 288)-net in base 32, using
- 5 times m-reduction [i] based on (25, 56, 288)-net in base 32, using
- base change [i] based on digital (9, 40, 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, 40, 288)-net over F128, using
- 5 times m-reduction [i] based on (25, 56, 288)-net in base 32, using
(51−25, 51, 559)-Net over F32 — Digital
Digital (26, 51, 559)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3251, 559, F32, 25) (dual of [559, 508, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3251, 1032, F32, 25) (dual of [1032, 981, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3249, 1024, F32, 25) (dual of [1024, 975, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3243, 1024, F32, 22) (dual of [1024, 981, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3251, 1032, F32, 25) (dual of [1032, 981, 26]-code), using
(51−25, 51, 318749)-Net in Base 32 — Upper bound on s
There is no (26, 51, 318750)-net in base 32, because
- 1 times m-reduction [i] would yield (26, 50, 318750)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1809 263973 196108 765136 441184 125608 617993 636938 433250 433853 670008 462856 718751 > 3250 [i]