Best Known (50, 61, s)-Nets in Base 7
(50, 61, 23543)-Net over F7 — Constructive and digital
Digital (50, 61, 23543)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 13)-net over F7, using
- 7 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (44, 55, 23530)-net over F7, using
- net defined by OOA [i] based on linear OOA(755, 23530, F7, 11, 11) (dual of [(23530, 11), 258775, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(755, 117651, F7, 11) (dual of [117651, 117596, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(755, 117655, F7, 11) (dual of [117655, 117600, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(755, 117649, F7, 11) (dual of [117649, 117594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(749, 117649, F7, 10) (dual of [117649, 117600, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(70, 6, F7, 0) (dual of [6, 6, 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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(755, 117655, F7, 11) (dual of [117655, 117600, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(755, 117651, F7, 11) (dual of [117651, 117596, 12]-code), using
- net defined by OOA [i] based on linear OOA(755, 23530, F7, 11, 11) (dual of [(23530, 11), 258775, 12]-NRT-code), using
- digital (1, 6, 13)-net over F7, using
(50, 61, 117679)-Net over F7 — Digital
Digital (50, 61, 117679)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(761, 117679, F7, 11) (dual of [117679, 117618, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(755, 117649, F7, 11) (dual of [117649, 117594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(731, 117649, F7, 6) (dual of [117649, 117618, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(76, 30, F7, 4) (dual of [30, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- 1 times truncation [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
(50, 61, large)-Net in Base 7 — Upper bound on s
There is no (50, 61, large)-net in base 7, because
- 9 times m-reduction [i] would yield (50, 52, large)-net in base 7, but