Best Known (109−24, 109, s)-Nets in Base 32
(109−24, 109, 87446)-Net over F32 — Constructive and digital
Digital (85, 109, 87446)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- digital (70, 94, 87382)-net over F32, using
- net defined by OOA [i] based on linear OOA(3294, 87382, F32, 24, 24) (dual of [(87382, 24), 2097074, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3294, 1048584, F32, 24) (dual of [1048584, 1048490, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3294, 1048585, F32, 24) (dual of [1048585, 1048491, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(3293, 1048576, F32, 24) (dual of [1048576, 1048483, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(3285, 1048576, F32, 22) (dual of [1048576, 1048491, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(321, 9, F32, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3294, 1048585, F32, 24) (dual of [1048585, 1048491, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3294, 1048584, F32, 24) (dual of [1048584, 1048490, 25]-code), using
- net defined by OOA [i] based on linear OOA(3294, 87382, F32, 24, 24) (dual of [(87382, 24), 2097074, 25]-NRT-code), using
- digital (3, 15, 64)-net over F32, using
(109−24, 109, 174765)-Net in Base 32 — Constructive
(85, 109, 174765)-net in base 32, using
- 321 times duplication [i] based on (84, 108, 174765)-net in base 32, using
- net defined by OOA [i] based on OOA(32108, 174765, S32, 24, 24), using
- OA 12-folding and stacking [i] based on OA(32108, 2097180, S32, 24), using
- discarding factors based on OA(32108, 2097183, S32, 24), using
- discarding parts of the base [i] based on linear OA(12877, 2097183, F128, 24) (dual of [2097183, 2097106, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- discarding parts of the base [i] based on linear OA(12877, 2097183, F128, 24) (dual of [2097183, 2097106, 25]-code), using
- discarding factors based on OA(32108, 2097183, S32, 24), using
- OA 12-folding and stacking [i] based on OA(32108, 2097180, S32, 24), using
- net defined by OOA [i] based on OOA(32108, 174765, S32, 24, 24), using
(109−24, 109, 4132357)-Net over F32 — Digital
Digital (85, 109, 4132357)-net over F32, using
(109−24, 109, large)-Net in Base 32 — Upper bound on s
There is no (85, 109, large)-net in base 32, because
- 22 times m-reduction [i] would yield (85, 87, large)-net in base 32, but