Best Known (57, 57+14, s)-Nets in Base 5
(57, 57+14, 2235)-Net over F5 — Constructive and digital
Digital (57, 71, 2235)-net over F5, using
- 51 times duplication [i] based on digital (56, 70, 2235)-net over F5, using
- net defined by OOA [i] based on linear OOA(570, 2235, F5, 14, 14) (dual of [(2235, 14), 31220, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(570, 15645, F5, 14) (dual of [15645, 15575, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(570, 15646, F5, 14) (dual of [15646, 15576, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(53, 21, F5, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(570, 15646, F5, 14) (dual of [15646, 15576, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(570, 15645, F5, 14) (dual of [15645, 15575, 15]-code), using
- net defined by OOA [i] based on linear OOA(570, 2235, F5, 14, 14) (dual of [(2235, 14), 31220, 15]-NRT-code), using
(57, 57+14, 15648)-Net over F5 — Digital
Digital (57, 71, 15648)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(571, 15648, F5, 14) (dual of [15648, 15577, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(570, 15646, F5, 14) (dual of [15646, 15576, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(53, 21, F5, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(570, 15647, F5, 13) (dual of [15647, 15577, 14]-code), using Gilbert–Varšamov bound and bm = 570 > Vbs−1(k−1) = 7 507126 497762 266353 326085 184394 983313 062616 834985 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(570, 15646, F5, 14) (dual of [15646, 15576, 15]-code), using
- construction X with Varšamov bound [i] based on
(57, 57+14, large)-Net in Base 5 — Upper bound on s
There is no (57, 71, large)-net in base 5, because
- 12 times m-reduction [i] would yield (57, 59, large)-net in base 5, but