Best Known (54, 54+10, s)-Nets in Base 32
(54, 54+10, 2202013)-Net over F32 — Constructive and digital
Digital (54, 64, 2202013)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (13, 18, 524293)-net over F32, using
- net defined by OOA [i] based on linear OOA(3218, 524293, F32, 5, 5) (dual of [(524293, 5), 2621447, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(3218, 1048587, F32, 5) (dual of [1048587, 1048569, 6]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(3217, 1048577, F32, 5) (dual of [1048577, 1048560, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(329, 1048577, F32, 3) (dual of [1048577, 1048568, 4]-code or 1048577-cap in PG(8,32)), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(329, 10, F32, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,32)), using
- dual of repetition code with length 10 [i]
- linear OA(321, 10, F32, 1) (dual of [10, 9, 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(3218, 1048587, F32, 5) (dual of [1048587, 1048569, 6]-code), using
- net defined by OOA [i] based on linear OOA(3218, 524293, F32, 5, 5) (dual of [(524293, 5), 2621447, 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 (13, 18, 524293)-net over F32, using
(54, 54+10, 2726297)-Net in Base 32 — Constructive
(54, 64, 2726297)-net in base 32, using
- (u, u+v)-construction [i] based on
- (14, 19, 1048577)-net in base 32, using
- net defined by OOA [i] based on OOA(3219, 1048577, S32, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(3219, 2097155, S32, 5), using
- discarding parts of the base [i] based on linear OA(12813, 2097155, F128, 5) (dual of [2097155, 2097142, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- discarding parts of the base [i] based on linear OA(12813, 2097155, F128, 5) (dual of [2097155, 2097142, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on OA(3219, 2097155, S32, 5), using
- net defined by OOA [i] based on OOA(3219, 1048577, S32, 5, 5), using
- (35, 45, 1677720)-net in base 32, using
- net defined by OOA [i] based on OOA(3245, 1677720, S32, 10, 10), using
- OA 5-folding and stacking [i] based on OA(3245, 8388600, S32, 10), using
- discarding factors based on OA(3245, large, S32, 10), using
- discarding parts of the base [i] 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 parts of the base [i] based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- discarding factors based on OA(3245, large, S32, 10), using
- OA 5-folding and stacking [i] based on OA(3245, 8388600, S32, 10), using
- net defined by OOA [i] based on OOA(3245, 1677720, S32, 10, 10), using
- (14, 19, 1048577)-net in base 32, using
(54, 54+10, large)-Net over F32 — Digital
Digital (54, 64, large)-net over F32, using
- t-expansion [i] based on digital (52, 64, large)-net over F32, using
- 2 times m-reduction [i] based on digital (52, 66, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3266, large, F32, 14) (dual of [large, large−66, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3266, large, F32, 14) (dual of [large, large−66, 15]-code), using
- 2 times m-reduction [i] based on digital (52, 66, large)-net over F32, using
(54, 54+10, large)-Net in Base 32 — Upper bound on s
There is no (54, 64, large)-net in base 32, because
- 8 times m-reduction [i] would yield (54, 56, large)-net in base 32, but