Best Known (79, 79+15, s)-Nets in Base 7
(79, 79+15, 117663)-Net over F7 — Constructive and digital
Digital (79, 94, 117663)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 13)-net over F7, using
- 5 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- 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
- net defined by OOA [i] based on linear OOA(786, 117650, F7, 15, 15) (dual of [(117650, 15), 1764664, 16]-NRT-code), using
- digital (1, 8, 13)-net over F7, using
(79, 79+15, 823589)-Net over F7 — Digital
Digital (79, 94, 823589)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(794, 823589, F7, 15) (dual of [823589, 823495, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(792, 823585, F7, 15) (dual of [823585, 823493, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(8) [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(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(77, 42, F7, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(14) ⊂ Ce(8) [i] based on
- linear OA(792, 823587, F7, 14) (dual of [823587, 823495, 15]-code), using Gilbert–Varšamov bound and bm = 792 > Vbs−1(k−1) = 168214 090390 102781 825678 277143 653289 297978 525048 332210 685074 534347 750109 618993 [i]
- linear OA(70, 2, F7, 0) (dual of [2, 2, 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(792, 823585, F7, 15) (dual of [823585, 823493, 16]-code), using
- construction X with Varšamov bound [i] based on
(79, 79+15, large)-Net in Base 7 — Upper bound on s
There is no (79, 94, large)-net in base 7, because
- 13 times m-reduction [i] would yield (79, 81, large)-net in base 7, but