Best Known (42, 42+9, s)-Nets in Base 7
(42, 42+9, 205887)-Net over F7 — Constructive and digital
Digital (42, 51, 205887)-net over F7, using
- 71 times duplication [i] based on digital (41, 50, 205887)-net over F7, using
- net defined by OOA [i] based on linear OOA(750, 205887, F7, 9, 9) (dual of [(205887, 9), 1852933, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(750, 823549, F7, 9) (dual of [823549, 823499, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(750, 823550, F7, 9) (dual of [823550, 823500, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- 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(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(70, 7, F7, 0) (dual of [7, 7, 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
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(750, 823550, F7, 9) (dual of [823550, 823500, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(750, 823549, F7, 9) (dual of [823549, 823499, 10]-code), using
- net defined by OOA [i] based on linear OOA(750, 205887, F7, 9, 9) (dual of [(205887, 9), 1852933, 10]-NRT-code), using
(42, 42+9, 612603)-Net over F7 — Digital
Digital (42, 51, 612603)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(751, 612603, F7, 9) (dual of [612603, 612552, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(751, 823552, F7, 9) (dual of [823552, 823501, 10]-code), using
- construction X4 applied to C([0,8]) ⊂ C([1,7]) [i] based on
- linear OA(750, 823542, F7, 9) (dual of [823542, 823492, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(742, 823542, F7, 7) (dual of [823542, 823500, 8]-code), using 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(79, 10, F7, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,7)), using
- dual of repetition code with length 10 [i]
- linear OA(71, 10, F7, 1) (dual of [10, 9, 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 C([0,8]) ⊂ C([1,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(751, 823552, F7, 9) (dual of [823552, 823501, 10]-code), using
(42, 42+9, large)-Net in Base 7 — Upper bound on s
There is no (42, 51, large)-net in base 7, because
- 7 times m-reduction [i] would yield (42, 44, large)-net in base 7, but