Best Known (60−10, 60, s)-Nets in Base 32
(60−10, 60, 1694108)-Net over F32 — Constructive and digital
Digital (50, 60, 1694108)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 14, 16388)-net over F32, using
- net defined by OOA [i] based on linear OOA(3214, 16388, F32, 5, 5) (dual of [(16388, 5), 81926, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(3214, 32777, F32, 5) (dual of [32777, 32763, 6]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(3213, 32769, F32, 5) (dual of [32769, 32756, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(327, 32769, F32, 3) (dual of [32769, 32762, 4]-code or 32769-cap in PG(6,32)), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(327, 8, F32, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,32)), using
- dual of repetition code with length 8 [i]
- linear OA(321, 8, F32, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(3214, 32777, F32, 5) (dual of [32777, 32763, 6]-code), using
- net defined by OOA [i] based on linear OOA(3214, 16388, F32, 5, 5) (dual of [(16388, 5), 81926, 6]-NRT-code), using
- digital (36, 46, 1677720)-net over F32, using
- net defined by OOA [i] based on linear OOA(3246, 1677720, F32, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3246, 8388600, F32, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, large, F32, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(3246, large, F32, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(3246, 8388600, F32, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(3246, 1677720, F32, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- digital (9, 14, 16388)-net over F32, using
(60−10, 60, 1808793)-Net in Base 32 — Constructive
(50, 60, 1808793)-net in base 32, using
- base change [i] based on digital (40, 50, 1808793)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (8, 13, 131073)-net over F64, using
- net defined by OOA [i] based on linear OOA(6413, 131073, F64, 5, 5) (dual of [(131073, 5), 655352, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(6413, 262147, F64, 5) (dual of [262147, 262134, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(6413, 262144, F64, 5) (dual of [262144, 262131, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(6410, 262144, F64, 4) (dual of [262144, 262134, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(6413, 262147, F64, 5) (dual of [262147, 262134, 6]-code), using
- net defined by OOA [i] based on linear OOA(6413, 131073, F64, 5, 5) (dual of [(131073, 5), 655352, 6]-NRT-code), using
- digital (27, 37, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- digital (8, 13, 131073)-net over F64, using
- (u, u+v)-construction [i] based on
(60−10, 60, large)-Net over F32 — Digital
Digital (50, 60, large)-net over F32, using
- t-expansion [i] based on digital (48, 60, large)-net over F32, using
- 1 times m-reduction [i] based on digital (48, 61, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- 1 times m-reduction [i] based on digital (48, 61, large)-net over F32, using
(60−10, 60, large)-Net in Base 32 — Upper bound on s
There is no (50, 60, large)-net in base 32, because
- 8 times m-reduction [i] would yield (50, 52, large)-net in base 32, but