Best Known (53−13, 53, s)-Nets in Base 49
(53−13, 53, 960804)-Net over F49 — Constructive and digital
Digital (40, 53, 960804)-net over F49, using
- net defined by OOA [i] based on linear OOA(4953, 960804, F49, 13, 13) (dual of [(960804, 13), 12490399, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4953, 5764825, F49, 13) (dual of [5764825, 5764772, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(4949, 5764801, F49, 13) (dual of [5764801, 5764752, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(494, 24, F49, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,49)), using
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- Reed–Solomon code RS(45,49) [i]
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(4953, 5764825, F49, 13) (dual of [5764825, 5764772, 14]-code), using
(53−13, 53, 5764825)-Net over F49 — Digital
Digital (40, 53, 5764825)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4953, 5764825, F49, 13) (dual of [5764825, 5764772, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(4949, 5764801, F49, 13) (dual of [5764801, 5764752, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(494, 24, F49, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,49)), using
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- Reed–Solomon code RS(45,49) [i]
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
(53−13, 53, large)-Net in Base 49 — Upper bound on s
There is no (40, 53, large)-net in base 49, because
- 11 times m-reduction [i] would yield (40, 42, large)-net in base 49, but