Best Known (31−16, 31, s)-Nets in Base 32
(31−16, 31, 131)-Net over F32 — Constructive and digital
Digital (15, 31, 131)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- 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 (0, 8, 33)-net over F32, using
(31−16, 31, 260)-Net in Base 32 — Constructive
(15, 31, 260)-net in base 32, using
- 1 times m-reduction [i] based on (15, 32, 260)-net in base 32, using
- base change [i] based on digital (3, 20, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- base change [i] based on digital (3, 20, 260)-net over F256, using
(31−16, 31, 411)-Net over F32 — Digital
Digital (15, 31, 411)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3231, 411, F32, 2, 16) (dual of [(411, 2), 791, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3231, 513, F32, 2, 16) (dual of [(513, 2), 995, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3231, 1026, F32, 16) (dual of [1026, 995, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(3231, 1024, F32, 16) (dual of [1024, 993, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3229, 1024, F32, 15) (dual of [1024, 995, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(15) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(3231, 1026, F32, 16) (dual of [1026, 995, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(3231, 513, F32, 2, 16) (dual of [(513, 2), 995, 17]-NRT-code), using
(31−16, 31, 82559)-Net in Base 32 — Upper bound on s
There is no (15, 31, 82560)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 45676 061176 804675 396653 550902 190643 773918 870033 > 3231 [i]