Best Known (65−31, 65, s)-Nets in Base 32
(65−31, 65, 218)-Net over F32 — Constructive and digital
Digital (34, 65, 218)-net over F32, using
- 1 times m-reduction [i] based on digital (34, 66, 218)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 23, 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, 43, 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, 23, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(65−31, 65, 288)-Net in Base 32 — Constructive
(34, 65, 288)-net in base 32, using
- t-expansion [i] based on (33, 65, 288)-net in base 32, using
- 19 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
- base change [i] based on digital (9, 60, 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, 60, 288)-net over F128, using
- 19 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
(65−31, 65, 776)-Net over F32 — Digital
Digital (34, 65, 776)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3265, 776, F32, 31) (dual of [776, 711, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3265, 1038, F32, 31) (dual of [1038, 973, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(3261, 1024, F32, 31) (dual of [1024, 963, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3251, 1024, F32, 26) (dual of [1024, 973, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(324, 14, F32, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3265, 1038, F32, 31) (dual of [1038, 973, 32]-code), using
(65−31, 65, 547485)-Net in Base 32 — Upper bound on s
There is no (34, 65, 547486)-net in base 32, because
- 1 times m-reduction [i] would yield (34, 64, 547486)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2 136043 615887 250514 358828 732617 913056 241950 429264 581032 922739 344678 916365 023118 134580 830709 204568 > 3264 [i]