Best Known (49, 49+11, s)-Nets in Base 7
(49, 49+11, 23538)-Net over F7 — Constructive and digital
Digital (49, 60, 23538)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence 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 (0, 5, 8)-net over F7, using
(49, 49+11, 117674)-Net over F7 — Digital
Digital (49, 60, 117674)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(760, 117674, F7, 11) (dual of [117674, 117614, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(758, 117670, F7, 11) (dual of [117670, 117612, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [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(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(73, 21, F7, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(758, 117672, F7, 10) (dual of [117672, 117614, 11]-code), using Gilbert–Varšamov bound and bm = 758 > Vbs−1(k−1) = 120086 969408 083632 508333 827333 714761 496963 234295 [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, 117670, F7, 11) (dual of [117670, 117612, 12]-code), using
- construction X with Varšamov bound [i] based on
(49, 49+11, large)-Net in Base 7 — Upper bound on s
There is no (49, 60, large)-net in base 7, because
- 9 times m-reduction [i] would yield (49, 51, large)-net in base 7, but