Best Known (44, 57, s)-Nets in Base 7
(44, 57, 2802)-Net over F7 — Constructive and digital
Digital (44, 57, 2802)-net over F7, using
- net defined by OOA [i] based on linear OOA(757, 2802, F7, 13, 13) (dual of [(2802, 13), 36369, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(757, 16813, F7, 13) (dual of [16813, 16756, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(757, 16818, F7, 13) (dual of [16818, 16761, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(756, 16807, F7, 13) (dual of [16807, 16751, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(757, 16818, F7, 13) (dual of [16818, 16761, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(757, 16813, F7, 13) (dual of [16813, 16756, 14]-code), using
(44, 57, 16406)-Net over F7 — Digital
Digital (44, 57, 16406)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(757, 16406, F7, 13) (dual of [16406, 16349, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(757, 16818, F7, 13) (dual of [16818, 16761, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(756, 16807, F7, 13) (dual of [16807, 16751, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(757, 16818, F7, 13) (dual of [16818, 16761, 14]-code), using
(44, 57, large)-Net in Base 7 — Upper bound on s
There is no (44, 57, large)-net in base 7, because
- 11 times m-reduction [i] would yield (44, 46, large)-net in base 7, but