Best Known (57−11, 57, s)-Nets in Base 7
(57−11, 57, 23532)-Net over F7 — Constructive and digital
Digital (46, 57, 23532)-net over F7, using
- 71 times duplication [i] based on digital (45, 56, 23532)-net over F7, using
- net defined by OOA [i] based on linear OOA(756, 23532, F7, 11, 11) (dual of [(23532, 11), 258796, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(756, 117661, F7, 11) (dual of [117661, 117605, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(756, 117662, F7, 11) (dual of [117662, 117606, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(755, 117649, F7, 11) (dual of [117649, 117594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(743, 117649, F7, 9) (dual of [117649, 117606, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(71, 13, F7, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(756, 117662, F7, 11) (dual of [117662, 117606, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(756, 117661, F7, 11) (dual of [117661, 117605, 12]-code), using
- net defined by OOA [i] based on linear OOA(756, 23532, F7, 11, 11) (dual of [(23532, 11), 258796, 12]-NRT-code), using
(57−11, 57, 117665)-Net over F7 — Digital
Digital (46, 57, 117665)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(757, 117665, F7, 11) (dual of [117665, 117608, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(756, 117663, F7, 11) (dual of [117663, 117607, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(755, 117649, F7, 11) (dual of [117649, 117594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(743, 117649, F7, 9) (dual of [117649, 117606, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(713, 14, F7, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,7)), using
- dual of repetition code with length 14 [i]
- linear OA(71, 14, F7, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(756, 117664, F7, 10) (dual of [117664, 117608, 11]-code), using Gilbert–Varšamov bound and bm = 756 > Vbs−1(k−1) = 120013 508722 005208 927485 790349 088061 237100 632759 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(756, 117663, F7, 11) (dual of [117663, 117607, 12]-code), using
- construction X with Varšamov bound [i] based on
(57−11, 57, large)-Net in Base 7 — Upper bound on s
There is no (46, 57, large)-net in base 7, because
- 9 times m-reduction [i] would yield (46, 48, large)-net in base 7, but