Best Known (29−5, 29, s)-Nets in Base 7
(29−5, 29, 411774)-Net over F7 — Constructive and digital
Digital (24, 29, 411774)-net over F7, using
- net defined by OOA [i] based on linear OOA(729, 411774, F7, 5, 5) (dual of [(411774, 5), 2058841, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(729, 823549, F7, 5) (dual of [823549, 823520, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(729, 823550, F7, 5) (dual of [823550, 823521, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(729, 823543, F7, 5) (dual of [823543, 823514, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(722, 823543, F7, 4) (dual of [823543, 823521, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(729, 823550, F7, 5) (dual of [823550, 823521, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(729, 823549, F7, 5) (dual of [823549, 823520, 6]-code), using
(29−5, 29, 823550)-Net over F7 — Digital
Digital (24, 29, 823550)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(729, 823550, F7, 5) (dual of [823550, 823521, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(729, 823543, F7, 5) (dual of [823543, 823514, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(722, 823543, F7, 4) (dual of [823543, 823521, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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(4) ⊂ Ce(3) [i] based on
(29−5, 29, large)-Net in Base 7 — Upper bound on s
There is no (24, 29, large)-net in base 7, because
- 3 times m-reduction [i] would yield (24, 26, large)-net in base 7, but