Best Known (12, 12+8, s)-Nets in Base 32
(12, 12+8, 1056)-Net over F32 — Constructive and digital
Digital (12, 20, 1056)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 33)-net over F32, using
- s-reduction based on digital (0, 0, s)-net over F32 with arbitrarily large s, using
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 0, 33)-net over F32 (see above)
- digital (0, 1, 33)-net over F32, using
- s-reduction based on digital (0, 1, s)-net over F32 with arbitrarily large s, using
- digital (0, 1, 33)-net over F32 (see above)
- digital (0, 1, 33)-net over F32 (see above)
- digital (0, 1, 33)-net over F32 (see above)
- digital (0, 2, 33)-net over F32, using
- digital (0, 2, 33)-net over F32 (see above)
- digital (0, 4, 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 (0, 8, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 0, 33)-net over F32, using
(12, 12+8, 2195)-Net over F32 — Digital
Digital (12, 20, 2195)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3220, 2195, F32, 8) (dual of [2195, 2175, 9]-code), using
- 1163 step Varšamov–Edel lengthening with (ri) = (2, 11 times 0, 1, 72 times 0, 1, 305 times 0, 1, 771 times 0) [i] based on linear OA(3215, 1027, F32, 8) (dual of [1027, 1012, 9]-code), using
- construction XX applied to C1 = C([1022,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([1022,6]) [i] based on
- linear OA(3213, 1023, F32, 7) (dual of [1023, 1010, 8]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(3213, 1023, F32, 7) (dual of [1023, 1010, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(3215, 1023, F32, 8) (dual of [1023, 1008, 9]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3211, 1023, F32, 6) (dual of [1023, 1012, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([1022,6]) [i] based on
- 1163 step Varšamov–Edel lengthening with (ri) = (2, 11 times 0, 1, 72 times 0, 1, 305 times 0, 1, 771 times 0) [i] based on linear OA(3215, 1027, F32, 8) (dual of [1027, 1012, 9]-code), using
(12, 12+8, 2395745)-Net in Base 32 — Upper bound on s
There is no (12, 20, 2395746)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 1 267651 287037 257134 102066 686216 > 3220 [i]