Best Known (30−8, 30, s)-Nets in Base 49
(30−8, 30, 1441202)-Net over F49 — Constructive and digital
Digital (22, 30, 1441202)-net over F49, using
- net defined by OOA [i] based on linear OOA(4930, 1441202, F49, 8, 8) (dual of [(1441202, 8), 11529586, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(4930, 5764808, F49, 8) (dual of [5764808, 5764778, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(4930, 5764810, F49, 8) (dual of [5764810, 5764780, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(4929, 5764801, F49, 8) (dual of [5764801, 5764772, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(4921, 5764801, F49, 6) (dual of [5764801, 5764780, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(491, 9, F49, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(4930, 5764810, F49, 8) (dual of [5764810, 5764780, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(4930, 5764808, F49, 8) (dual of [5764808, 5764778, 9]-code), using
(30−8, 30, 5764811)-Net over F49 — Digital
Digital (22, 30, 5764811)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4930, 5764811, F49, 8) (dual of [5764811, 5764781, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(4929, 5764801, F49, 8) (dual of [5764801, 5764772, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(4921, 5764801, F49, 6) (dual of [5764801, 5764780, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(499, 10, F49, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,49)), using
- dual of repetition code with length 10 [i]
- linear OA(491, 10, F49, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, 49, F49, 1) (dual of [49, 48, 2]-code), using
- Reed–Solomon code RS(48,49) [i]
- discarding factors / shortening the dual code based on linear OA(491, 49, F49, 1) (dual of [49, 48, 2]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
(30−8, 30, large)-Net in Base 49 — Upper bound on s
There is no (22, 30, large)-net in base 49, because
- 6 times m-reduction [i] would yield (22, 24, large)-net in base 49, but