Best Known (68−13, 68, s)-Nets in Base 7
(68−13, 68, 19610)-Net over F7 — Constructive and digital
Digital (55, 68, 19610)-net over F7, using
- net defined by OOA [i] based on linear OOA(768, 19610, F7, 13, 13) (dual of [(19610, 13), 254862, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(768, 117661, F7, 13) (dual of [117661, 117593, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(768, 117662, F7, 13) (dual of [117662, 117594, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(768, 117662, F7, 13) (dual of [117662, 117594, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(768, 117661, F7, 13) (dual of [117661, 117593, 14]-code), using
(68−13, 68, 114883)-Net over F7 — Digital
Digital (55, 68, 114883)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(768, 114883, F7, 13) (dual of [114883, 114815, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(768, 117662, F7, 13) (dual of [117662, 117594, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(768, 117662, F7, 13) (dual of [117662, 117594, 14]-code), using
(68−13, 68, large)-Net in Base 7 — Upper bound on s
There is no (55, 68, large)-net in base 7, because
- 11 times m-reduction [i] would yield (55, 57, large)-net in base 7, but