Best Known (25, 29, s)-Nets in Base 7
(25, 29, 5764861)-Net over F7 — Constructive and digital
Digital (25, 29, 5764861)-net over F7, using
- net defined by OOA [i] based on linear OOA(729, 5764861, F7, 4, 4) (dual of [(5764861, 4), 23059415, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(729, 5764861, F7, 3, 4) (dual of [(5764861, 3), 17294554, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(73, 57, F7, 3, 2) (dual of [(57, 3), 168, 3]-NRT-code), using
- appending 1 arbitrary column [i] based on linear OOA(73, 57, F7, 2, 2) (dual of [(57, 2), 111, 3]-NRT-code), using
- appending kth column [i] based on linear OA(73, 57, F7, 2) (dual of [57, 54, 3]-code), using
- Hamming code H(3,7) [i]
- appending kth column [i] based on linear OA(73, 57, F7, 2) (dual of [57, 54, 3]-code), using
- appending 1 arbitrary column [i] based on linear OOA(73, 57, F7, 2, 2) (dual of [(57, 2), 111, 3]-NRT-code), using
- linear OOA(726, 5764804, F7, 3, 4) (dual of [(5764804, 3), 17294386, 5]-NRT-code), using
- trace code [i] based on linear OOA(4913, 2882402, F49, 3, 4) (dual of [(2882402, 3), 8647193, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(4913, 5764804, F49, 4) (dual of [5764804, 5764791, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(4913, 5764805, F49, 4) (dual of [5764805, 5764792, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(4913, 5764801, F49, 4) (dual of [5764801, 5764788, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(499, 5764801, F49, 3) (dual of [5764801, 5764792, 4]-code or 5764801-cap in PG(8,49)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(4913, 5764805, F49, 4) (dual of [5764805, 5764792, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(4913, 5764804, F49, 4) (dual of [5764804, 5764791, 5]-code), using
- trace code [i] based on linear OOA(4913, 2882402, F49, 3, 4) (dual of [(2882402, 3), 8647193, 5]-NRT-code), using
- linear OOA(73, 57, F7, 3, 2) (dual of [(57, 3), 168, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(729, 5764861, F7, 3, 4) (dual of [(5764861, 3), 17294554, 5]-NRT-code), using
(25, 29, large)-Net over F7 — Digital
Digital (25, 29, large)-net over F7, using
- 73 times duplication [i] based on digital (22, 26, large)-net over F7, using
- net defined by OOA [i] based on linear OOA(726, large, F7, 4, 4), using
- appending kth column [i] based on linear OOA(726, large, F7, 3, 4), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(726, large, F7, 4) (dual of [large, large−26, 5]-code), using
- trace code [i] based on linear OA(4913, 5764801, F49, 4) (dual of [5764801, 5764788, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- trace code [i] based on linear OA(4913, 5764801, F49, 4) (dual of [5764801, 5764788, 5]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(726, large, F7, 4) (dual of [large, large−26, 5]-code), using
- appending kth column [i] based on linear OOA(726, large, F7, 3, 4), using
- net defined by OOA [i] based on linear OOA(726, large, F7, 4, 4), using
(25, 29, large)-Net in Base 7 — Upper bound on s
There is no (25, 29, large)-net in base 7, because
- 2 times m-reduction [i] would yield (25, 27, large)-net in base 7, but