Best Known (53−13, 53, s)-Nets in Base 32
(53−13, 53, 174766)-Net over F32 — Constructive and digital
Digital (40, 53, 174766)-net over F32, using
- net defined by OOA [i] based on linear OOA(3253, 174766, F32, 13, 13) (dual of [(174766, 13), 2271905, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3253, 1048597, F32, 13) (dual of [1048597, 1048544, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3253, 1048600, F32, 13) (dual of [1048600, 1048547, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(3249, 1048576, F32, 13) (dual of [1048576, 1048527, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3229, 1048576, F32, 8) (dual of [1048576, 1048547, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(324, 24, F32, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(3253, 1048600, F32, 13) (dual of [1048600, 1048547, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3253, 1048597, F32, 13) (dual of [1048597, 1048544, 14]-code), using
(53−13, 53, 349525)-Net in Base 32 — Constructive
(40, 53, 349525)-net in base 32, using
- 321 times duplication [i] based on (39, 52, 349525)-net in base 32, using
- net defined by OOA [i] based on OOA(3252, 349525, S32, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3252, 2097151, S32, 13), using
- discarding factors based on OA(3252, 2097155, S32, 13), using
- discarding parts of the base [i] based on linear OA(12837, 2097155, F128, 13) (dual of [2097155, 2097118, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12837, 2097155, F128, 13) (dual of [2097155, 2097118, 14]-code), using
- discarding factors based on OA(3252, 2097155, S32, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3252, 2097151, S32, 13), using
- net defined by OOA [i] based on OOA(3252, 349525, S32, 13, 13), using
(53−13, 53, 1048600)-Net over F32 — Digital
Digital (40, 53, 1048600)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3253, 1048600, F32, 13) (dual of [1048600, 1048547, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(3249, 1048576, F32, 13) (dual of [1048576, 1048527, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3229, 1048576, F32, 8) (dual of [1048576, 1048547, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(324, 24, F32, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
(53−13, 53, large)-Net in Base 32 — Upper bound on s
There is no (40, 53, large)-net in base 32, because
- 11 times m-reduction [i] would yield (40, 42, large)-net in base 32, but