Best Known (70−13, 70, s)-Nets in Base 32
(70−13, 70, 1398166)-Net over F32 — Constructive and digital
Digital (57, 70, 1398166)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 66)-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 (0, 6, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 3, 33)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (48, 61, 1398100)-net over F32, using
- net defined by OOA [i] based on linear OOA(3261, 1398100, F32, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3261, 8388601, F32, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3261, 8388601, F32, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(3261, 1398100, F32, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (3, 9, 66)-net over F32, using
(70−13, 70, 1398442)-Net in Base 32 — Constructive
(57, 70, 1398442)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (5, 11, 342)-net over F32, using
- net defined by OOA [i] based on linear OOA(3211, 342, F32, 6, 6) (dual of [(342, 6), 2041, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(3211, 1026, F32, 6) (dual of [1026, 1015, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(3211, 1024, F32, 6) (dual of [1024, 1013, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(329, 1024, F32, 5) (dual of [1024, 1015, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(3211, 1026, F32, 6) (dual of [1026, 1015, 7]-code), using
- net defined by OOA [i] based on linear OOA(3211, 342, F32, 6, 6) (dual of [(342, 6), 2041, 7]-NRT-code), using
- (46, 59, 1398100)-net in base 32, using
- net defined by OOA [i] based on OOA(3259, 1398100, S32, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3259, 8388601, S32, 13), using
- discarding factors based on OA(3259, large, S32, 13), using
- discarding parts of the base [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding parts of the base [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- discarding factors based on OA(3259, large, S32, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3259, 8388601, S32, 13), using
- net defined by OOA [i] based on OOA(3259, 1398100, S32, 13, 13), using
- digital (5, 11, 342)-net over F32, using
(70−13, 70, large)-Net over F32 — Digital
Digital (57, 70, large)-net over F32, using
- t-expansion [i] based on digital (56, 70, large)-net over F32, using
- 1 times m-reduction [i] based on digital (56, 71, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3271, large, F32, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3271, large, F32, 15) (dual of [large, large−71, 16]-code), using
- 1 times m-reduction [i] based on digital (56, 71, large)-net over F32, using
(70−13, 70, large)-Net in Base 32 — Upper bound on s
There is no (57, 70, large)-net in base 32, because
- 11 times m-reduction [i] would yield (57, 59, large)-net in base 32, but