Best Known (67, 67+13, s)-Nets in Base 7
(67, 67+13, 137259)-Net over F7 — Constructive and digital
Digital (67, 80, 137259)-net over F7, using
- 71 times duplication [i] based on digital (66, 79, 137259)-net over F7, using
- net defined by OOA [i] based on linear OOA(779, 137259, F7, 13, 13) (dual of [(137259, 13), 1784288, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(779, 823555, F7, 13) (dual of [823555, 823476, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(779, 823558, F7, 13) (dual of [823558, 823479, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- 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(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(71, 15, F7, 1) (dual of [15, 14, 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(779, 823558, F7, 13) (dual of [823558, 823479, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(779, 823555, F7, 13) (dual of [823555, 823476, 14]-code), using
- net defined by OOA [i] based on linear OOA(779, 137259, F7, 13, 13) (dual of [(137259, 13), 1784288, 14]-NRT-code), using
(67, 67+13, 823561)-Net over F7 — Digital
Digital (67, 80, 823561)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(780, 823561, F7, 13) (dual of [823561, 823481, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(779, 823559, F7, 13) (dual of [823559, 823480, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 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(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(715, 16, F7, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,7)), using
- dual of repetition code with length 16 [i]
- linear OA(71, 16, F7, 1) (dual of [16, 15, 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(12) ⊂ Ce(10) [i] based on
- linear OA(779, 823560, F7, 12) (dual of [823560, 823481, 13]-code), using Gilbert–Varšamov bound and bm = 779 > Vbs−1(k−1) = 1 074290 889782 162311 524754 763248 220837 953076 332077 124906 240848 722167 [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(779, 823559, F7, 13) (dual of [823559, 823480, 14]-code), using
- construction X with Varšamov bound [i] based on
(67, 67+13, large)-Net in Base 7 — Upper bound on s
There is no (67, 80, large)-net in base 7, because
- 11 times m-reduction [i] would yield (67, 69, large)-net in base 7, but