Best Known (55−11, 55, s)-Nets in Base 32
(55−11, 55, 1677720)-Net over F32 — Constructive and digital
Digital (44, 55, 1677720)-net over F32, using
- 324 times duplication [i] based on digital (40, 51, 1677720)-net over F32, using
- net defined by OOA [i] based on linear OOA(3251, 1677720, F32, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3251, 8388601, F32, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3251, large, F32, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3251, large, F32, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3251, 8388601, F32, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(3251, 1677720, F32, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
(55−11, 55, 1677753)-Net in Base 32 — Constructive
(44, 55, 1677753)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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
- (39, 50, 1677720)-net in base 32, using
- net defined by OOA [i] based on OOA(3250, 1677720, S32, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(3250, 8388601, S32, 11), using
- discarding factors based on OA(3250, large, S32, 11), using
- discarding parts of the base [i] based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding parts of the base [i] based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- discarding factors based on OA(3250, large, S32, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(3250, 8388601, S32, 11), using
- net defined by OOA [i] based on OOA(3250, 1677720, S32, 11, 11), using
- digital (0, 5, 33)-net over F32, using
(55−11, 55, large)-Net over F32 — Digital
Digital (44, 55, large)-net over F32, using
- 1 times m-reduction [i] based on digital (44, 56, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3256, large, F32, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3256, large, F32, 12) (dual of [large, large−56, 13]-code), using
(55−11, 55, large)-Net in Base 32 — Upper bound on s
There is no (44, 55, large)-net in base 32, because
- 9 times m-reduction [i] would yield (44, 46, large)-net in base 32, but