Best Known (50−10, 50, s)-Nets in Base 7
(50−10, 50, 23532)-Net over F7 — Constructive and digital
Digital (40, 50, 23532)-net over F7, using
- net defined by OOA [i] based on linear OOA(750, 23532, F7, 10, 10) (dual of [(23532, 10), 235270, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(750, 117660, F7, 10) (dual of [117660, 117610, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(750, 117662, F7, 10) (dual of [117662, 117612, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(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(71, 13, F7, 1) (dual of [13, 12, 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(750, 117662, F7, 10) (dual of [117662, 117612, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(750, 117660, F7, 10) (dual of [117660, 117610, 11]-code), using
(50−10, 50, 94134)-Net over F7 — Digital
Digital (40, 50, 94134)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(750, 94134, F7, 10) (dual of [94134, 94084, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(750, 117662, F7, 10) (dual of [117662, 117612, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(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(71, 13, F7, 1) (dual of [13, 12, 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(750, 117662, F7, 10) (dual of [117662, 117612, 11]-code), using
(50−10, 50, large)-Net in Base 7 — Upper bound on s
There is no (40, 50, large)-net in base 7, because
- 8 times m-reduction [i] would yield (40, 42, large)-net in base 7, but