Best Known (56−11, 56, s)-Nets in Base 7
(56−11, 56, 23532)-Net over F7 — Constructive and digital
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
(56−11, 56, 100941)-Net over F7 — Digital
Digital (45, 56, 100941)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(756, 100941, F7, 11) (dual of [100941, 100885, 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
(56−11, 56, large)-Net in Base 7 — Upper bound on s
There is no (45, 56, large)-net in base 7, because
- 9 times m-reduction [i] would yield (45, 47, large)-net in base 7, but