Best Known (18, 18+13, s)-Nets in Base 32
(18, 18+13, 231)-Net over F32 — Constructive and digital
Digital (18, 31, 231)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- 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, 2, 33)-net over F32, using
- digital (0, 2, 33)-net over F32 (see above)
- digital (0, 3, 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, 4, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 6, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 13, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 1, 33)-net over F32, using
(18, 18+13, 683)-Net in Base 32 — Constructive
(18, 31, 683)-net in base 32, using
- net defined by OOA [i] based on OOA(3231, 683, S32, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3231, 4099, S32, 13), using
- 1 times code embedding in larger space [i] based on OA(3230, 4098, S32, 13), using
- discarding parts of the base [i] based on linear OA(6425, 4098, F64, 13) (dual of [4098, 4073, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(6425, 4098, F64, 13) (dual of [4098, 4073, 14]-code), using
- 1 times code embedding in larger space [i] based on OA(3230, 4098, S32, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3231, 4099, S32, 13), using
(18, 18+13, 1373)-Net over F32 — Digital
Digital (18, 31, 1373)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3231, 1373, F32, 13) (dual of [1373, 1342, 14]-code), using
- 340 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 25 times 0, 1, 87 times 0, 1, 219 times 0) [i] based on linear OA(3225, 1027, F32, 13) (dual of [1027, 1002, 14]-code), using
- construction XX applied to C1 = C([1022,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([1022,11]) [i] based on
- linear OA(3223, 1023, F32, 12) (dual of [1023, 1000, 13]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3223, 1023, F32, 12) (dual of [1023, 1000, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3225, 1023, F32, 13) (dual of [1023, 998, 14]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3221, 1023, F32, 11) (dual of [1023, 1002, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [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([1022,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([1022,11]) [i] based on
- 340 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 25 times 0, 1, 87 times 0, 1, 219 times 0) [i] based on linear OA(3225, 1027, F32, 13) (dual of [1027, 1002, 14]-code), using
(18, 18+13, 3240484)-Net in Base 32 — Upper bound on s
There is no (18, 31, 3240485)-net in base 32, because
- 1 times m-reduction [i] would yield (18, 30, 3240485)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1427 249584 888294 794553 828694 716444 850621 442448 > 3230 [i]