Best Known (68−20, 68, s)-Nets in Base 32
(68−20, 68, 3310)-Net over F32 — Constructive and digital
Digital (48, 68, 3310)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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 (38, 58, 3277)-net over F32, using
- net defined by OOA [i] based on linear OOA(3258, 3277, F32, 20, 20) (dual of [(3277, 20), 65482, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3258, 32770, F32, 20) (dual of [32770, 32712, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3258, 32771, F32, 20) (dual of [32771, 32713, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3258, 32768, F32, 20) (dual of [32768, 32710, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3255, 32768, F32, 19) (dual of [32768, 32713, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(320, 3, F32, 0) (dual of [3, 3, 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(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3258, 32771, F32, 20) (dual of [32771, 32713, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3258, 32770, F32, 20) (dual of [32770, 32712, 21]-code), using
- net defined by OOA [i] based on linear OOA(3258, 3277, F32, 20, 20) (dual of [(3277, 20), 65482, 21]-NRT-code), using
- digital (0, 10, 33)-net over F32, using
(68−20, 68, 6554)-Net in Base 32 — Constructive
(48, 68, 6554)-net in base 32, using
- t-expansion [i] based on (47, 68, 6554)-net in base 32, using
- net defined by OOA [i] based on OOA(3268, 6554, S32, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(3268, 65541, S32, 21), using
- discarding factors based on OA(3268, 65542, S32, 21), using
- discarding parts of the base [i] based on linear OA(25642, 65542, F256, 21) (dual of [65542, 65500, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding parts of the base [i] based on linear OA(25642, 65542, F256, 21) (dual of [65542, 65500, 22]-code), using
- discarding factors based on OA(3268, 65542, S32, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(3268, 65541, S32, 21), using
- net defined by OOA [i] based on OOA(3268, 6554, S32, 21, 21), using
(68−20, 68, 62341)-Net over F32 — Digital
Digital (48, 68, 62341)-net over F32, using
(68−20, 68, large)-Net in Base 32 — Upper bound on s
There is no (48, 68, large)-net in base 32, because
- 18 times m-reduction [i] would yield (48, 50, large)-net in base 32, but