Best Known (56, 56+10, s)-Nets in Base 7
(56, 56+10, 1152963)-Net over F7 — Constructive and digital
Digital (56, 66, 1152963)-net over F7, using
- net defined by OOA [i] based on linear OOA(766, 1152963, F7, 10, 10) (dual of [(1152963, 10), 11529564, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(766, 5764815, F7, 10) (dual of [5764815, 5764749, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(766, 5764818, F7, 10) (dual of [5764818, 5764752, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(71, 17, F7, 1) (dual of [17, 16, 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(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(766, 5764818, F7, 10) (dual of [5764818, 5764752, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(766, 5764815, F7, 10) (dual of [5764815, 5764749, 11]-code), using
(56, 56+10, 4612751)-Net over F7 — Digital
Digital (56, 66, 4612751)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(766, 4612751, F7, 10) (dual of [4612751, 4612685, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(766, 5764818, F7, 10) (dual of [5764818, 5764752, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(71, 17, F7, 1) (dual of [17, 16, 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(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(766, 5764818, F7, 10) (dual of [5764818, 5764752, 11]-code), using
(56, 56+10, large)-Net in Base 7 — Upper bound on s
There is no (56, 66, large)-net in base 7, because
- 8 times m-reduction [i] would yield (56, 58, large)-net in base 7, but