Best Known (28, 28+5, s)-Nets in Base 7
(28, 28+5, 2882404)-Net over F7 — Constructive and digital
Digital (28, 33, 2882404)-net over F7, using
- net defined by OOA [i] based on linear OOA(733, 2882404, F7, 5, 5) (dual of [(2882404, 5), 14411987, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(733, 5764809, F7, 5) (dual of [5764809, 5764776, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(733, 5764801, F7, 5) (dual of [5764801, 5764768, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(725, 5764801, F7, 4) (dual of [5764801, 5764776, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 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
- OOA 2-folding and stacking with additional row [i] based on linear OA(733, 5764809, F7, 5) (dual of [5764809, 5764776, 6]-code), using
(28, 28+5, 5764809)-Net over F7 — Digital
Digital (28, 33, 5764809)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(733, 5764809, F7, 5) (dual of [5764809, 5764776, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(733, 5764801, F7, 5) (dual of [5764801, 5764768, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(725, 5764801, F7, 4) (dual of [5764801, 5764776, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 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
(28, 28+5, large)-Net in Base 7 — Upper bound on s
There is no (28, 33, large)-net in base 7, because
- 3 times m-reduction [i] would yield (28, 30, large)-net in base 7, but