Best Known (37, 37+31, s)-Nets in Base 32
(37, 37+31, 240)-Net over F32 — Constructive and digital
Digital (37, 68, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 26, 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, 42, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 26, 120)-net over F32, using
(37, 37+31, 306)-Net in Base 32 — Constructive
(37, 68, 306)-net in base 32, using
- 321 times duplication [i] based on (36, 67, 306)-net in base 32, using
- (u, u+v)-construction [i] based on
- (6, 21, 129)-net in base 32, using
- base change [i] based on digital (0, 15, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 15, 129)-net over F128, using
- (15, 46, 177)-net in base 32, using
- 2 times m-reduction [i] based on (15, 48, 177)-net in base 32, using
- base change [i] based on digital (7, 40, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- base change [i] based on digital (7, 40, 177)-net over F64, using
- 2 times m-reduction [i] based on (15, 48, 177)-net in base 32, using
- (6, 21, 129)-net in base 32, using
- (u, u+v)-construction [i] based on
(37, 37+31, 1094)-Net over F32 — Digital
Digital (37, 68, 1094)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3268, 1094, F32, 31) (dual of [1094, 1026, 32]-code), using
- 60 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 6 times 0, 1, 15 times 0, 1, 34 times 0) [i] based on linear OA(3261, 1027, F32, 31) (dual of [1027, 966, 32]-code), using
- construction XX applied to C1 = C([1022,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([1022,29]) [i] based on
- linear OA(3259, 1023, F32, 30) (dual of [1023, 964, 31]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,28}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3259, 1023, F32, 30) (dual of [1023, 964, 31]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3261, 1023, F32, 31) (dual of [1023, 962, 32]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,29}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3257, 1023, F32, 29) (dual of [1023, 966, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [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([1022,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([1022,29]) [i] based on
- 60 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 6 times 0, 1, 15 times 0, 1, 34 times 0) [i] based on linear OA(3261, 1027, F32, 31) (dual of [1027, 966, 32]-code), using
(37, 37+31, 1094977)-Net in Base 32 — Upper bound on s
There is no (37, 68, 1094978)-net in base 32, because
- 1 times m-reduction [i] would yield (37, 67, 1094978)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 69992 454470 147879 087699 555747 957954 165069 799289 768754 679362 619778 124127 357581 736466 223791 105744 247968 > 3267 [i]