Best Known (50, 50+10, s)-Nets in Base 7
(50, 50+10, 164712)-Net over F7 — Constructive and digital
Digital (50, 60, 164712)-net over F7, using
- 71 times duplication [i] based on digital (49, 59, 164712)-net over F7, using
- net defined by OOA [i] based on linear OOA(759, 164712, F7, 10, 10) (dual of [(164712, 10), 1647061, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(759, 823560, F7, 10) (dual of [823560, 823501, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(758, 823559, F7, 10) (dual of [823559, 823501, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(758, 823559, F7, 10) (dual of [823559, 823501, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(759, 823560, F7, 10) (dual of [823560, 823501, 11]-code), using
- net defined by OOA [i] based on linear OOA(759, 164712, F7, 10, 10) (dual of [(164712, 10), 1647061, 11]-NRT-code), using
(50, 50+10, 823563)-Net over F7 — Digital
Digital (50, 60, 823563)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(760, 823563, F7, 10) (dual of [823563, 823503, 11]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(758, 823559, F7, 10) (dual of [823559, 823501, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- linear OA(758, 823561, F7, 9) (dual of [823561, 823503, 10]-code), using Gilbert–Varšamov bound and bm = 758 > Vbs−1(k−1) = 8 815305 684839 634022 561157 651765 352248 770272 775873 [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(758, 823559, F7, 10) (dual of [823559, 823501, 11]-code), using
- construction X with Varšamov bound [i] based on
(50, 50+10, large)-Net in Base 7 — Upper bound on s
There is no (50, 60, large)-net in base 7, because
- 8 times m-reduction [i] would yield (50, 52, large)-net in base 7, but