Best Known (18, 18+6, s)-Nets in Base 32
(18, 18+6, 349559)-Net over F32 — Constructive and digital
Digital (18, 24, 349559)-net over F32, using
- (u, u+v)-construction [i] based on
- 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 (15, 21, 349526)-net over F32, using
- net defined by OOA [i] based on linear OOA(3221, 349526, F32, 6, 6) (dual of [(349526, 6), 2097135, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(3221, 1048578, F32, 6) (dual of [1048578, 1048557, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 1048580, F32, 6) (dual of [1048580, 1048559, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(3221, 1048576, F32, 6) (dual of [1048576, 1048555, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(3217, 1048576, F32, 5) (dual of [1048576, 1048559, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(3221, 1048580, F32, 6) (dual of [1048580, 1048559, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(3221, 1048578, F32, 6) (dual of [1048578, 1048557, 7]-code), using
- net defined by OOA [i] based on linear OOA(3221, 349526, F32, 6, 6) (dual of [(349526, 6), 2097135, 7]-NRT-code), using
- digital (0, 3, 33)-net over F32, using
(18, 18+6, 699053)-Net in Base 32 — Constructive
(18, 24, 699053)-net in base 32, using
- net defined by OOA [i] based on OOA(3224, 699053, S32, 6, 6), using
- OA 3-folding and stacking [i] based on OA(3224, 2097159, S32, 6), using
- discarding factors based on OA(3224, 2097160, S32, 6), using
- discarding parts of the base [i] based on linear OA(12817, 2097160, F128, 6) (dual of [2097160, 2097143, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 8, F128, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,128)), using
- dual of repetition code with length 8 [i]
- linear OA(1281, 8, F128, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- Reed–Solomon code RS(127,128) [i]
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- discarding parts of the base [i] based on linear OA(12817, 2097160, F128, 6) (dual of [2097160, 2097143, 7]-code), using
- discarding factors based on OA(3224, 2097160, S32, 6), using
- OA 3-folding and stacking [i] based on OA(3224, 2097159, S32, 6), using
(18, 18+6, 1409922)-Net over F32 — Digital
Digital (18, 24, 1409922)-net over F32, using
(18, 18+6, 2097159)-Net in Base 32
(18, 24, 2097159)-net in base 32, using
- base change [i] based on (14, 20, 2097159)-net in base 64, using
- net defined by OOA [i] based on OOA(6420, 2097159, S64, 9, 6), using
- OOA stacking with additional row [i] based on OOA(6420, 2097160, S64, 3, 6), using
- discarding parts of the base [i] based on linear OOA(12817, 2097160, F128, 3, 6) (dual of [(2097160, 3), 6291463, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12817, 2097160, F128, 6) (dual of [2097160, 2097143, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 8, F128, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,128)), using
- dual of repetition code with length 8 [i]
- linear OA(1281, 8, F128, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- Reed–Solomon code RS(127,128) [i]
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12817, 2097160, F128, 6) (dual of [2097160, 2097143, 7]-code), using
- discarding parts of the base [i] based on linear OOA(12817, 2097160, F128, 3, 6) (dual of [(2097160, 3), 6291463, 7]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(6420, 2097160, S64, 3, 6), using
- net defined by OOA [i] based on OOA(6420, 2097159, S64, 9, 6), using
(18, 18+6, large)-Net in Base 32 — Upper bound on s
There is no (18, 24, large)-net in base 32, because
- 4 times m-reduction [i] would yield (18, 20, large)-net in base 32, but