Best Known (86−15, 86, s)-Nets in Base 7
(86−15, 86, 117650)-Net over F7 — Constructive and digital
Digital (71, 86, 117650)-net over F7, using
- net defined by OOA [i] based on linear OOA(786, 117650, F7, 15, 15) (dual of [(117650, 15), 1764664, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(786, 823551, F7, 15) (dual of [823551, 823465, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(778, 823543, F7, 13) (dual of [823543, 823465, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(71, 8, F7, 1) (dual of [8, 7, 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(14) ⊂ Ce(12) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(786, 823551, F7, 15) (dual of [823551, 823465, 16]-code), using
(86−15, 86, 411775)-Net over F7 — Digital
Digital (71, 86, 411775)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(786, 411775, F7, 2, 15) (dual of [(411775, 2), 823464, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(786, 823550, F7, 15) (dual of [823550, 823464, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(786, 823551, F7, 15) (dual of [823551, 823465, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(778, 823543, F7, 13) (dual of [823543, 823465, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(71, 8, F7, 1) (dual of [8, 7, 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(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(786, 823551, F7, 15) (dual of [823551, 823465, 16]-code), using
- OOA 2-folding [i] based on linear OA(786, 823550, F7, 15) (dual of [823550, 823464, 16]-code), using
(86−15, 86, large)-Net in Base 7 — Upper bound on s
There is no (71, 86, large)-net in base 7, because
- 13 times m-reduction [i] would yield (71, 73, large)-net in base 7, but