Best Known (69−20, 69, s)-Nets in Base 32
(69−20, 69, 3321)-Net over F32 — Constructive and digital
Digital (49, 69, 3321)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-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 (1, 11, 44)-net over F32, using
(69−20, 69, 6555)-Net in Base 32 — Constructive
(49, 69, 6555)-net in base 32, using
- net defined by OOA [i] based on OOA(3269, 6555, S32, 20, 20), using
- OA 10-folding and stacking [i] based on OA(3269, 65550, S32, 20), using
- discarding parts of the base [i] based on linear OA(25643, 65550, F256, 20) (dual of [65550, 65507, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(25629, 65536, F256, 15) (dual of [65536, 65507, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- discarding parts of the base [i] based on linear OA(25643, 65550, F256, 20) (dual of [65550, 65507, 21]-code), using
- OA 10-folding and stacking [i] based on OA(3269, 65550, S32, 20), using
(69−20, 69, 74813)-Net over F32 — Digital
Digital (49, 69, 74813)-net over F32, using
(69−20, 69, large)-Net in Base 32 — Upper bound on s
There is no (49, 69, large)-net in base 32, because
- 18 times m-reduction [i] would yield (49, 51, large)-net in base 32, but