Best Known (23−10, 23, s)-Nets in Base 49
(23−10, 23, 483)-Net over F49 — Constructive and digital
Digital (13, 23, 483)-net over F49, using
- net defined by OOA [i] based on linear OOA(4923, 483, F49, 10, 10) (dual of [(483, 10), 4807, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(4923, 2415, F49, 10) (dual of [2415, 2392, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(4919, 2401, F49, 10) (dual of [2401, 2382, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(499, 2401, F49, 5) (dual of [2401, 2392, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(494, 14, F49, 4) (dual of [14, 10, 5]-code or 14-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(9) ⊂ Ce(4) [i] based on
- OA 5-folding and stacking [i] based on linear OA(4923, 2415, F49, 10) (dual of [2415, 2392, 11]-code), using
(23−10, 23, 2509)-Net over F49 — Digital
Digital (13, 23, 2509)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4923, 2509, F49, 10) (dual of [2509, 2486, 11]-code), using
- 102 step Varšamov–Edel lengthening with (ri) = (2, 1, 12 times 0, 1, 87 times 0) [i] based on linear OA(4919, 2403, F49, 10) (dual of [2403, 2384, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(4919, 2401, F49, 10) (dual of [2401, 2382, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(4917, 2401, F49, 9) (dual of [2401, 2384, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- 102 step Varšamov–Edel lengthening with (ri) = (2, 1, 12 times 0, 1, 87 times 0) [i] based on linear OA(4919, 2403, F49, 10) (dual of [2403, 2384, 11]-code), using
(23−10, 23, 3232188)-Net in Base 49 — Upper bound on s
There is no (13, 23, 3232189)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 749 048767 967633 259659 373487 314330 652465 > 4923 [i]