Best Known (18, 18+14, s)-Nets in Base 49
(18, 18+14, 345)-Net over F49 — Constructive and digital
Digital (18, 32, 345)-net over F49, using
- 491 times duplication [i] based on digital (17, 31, 345)-net over F49, using
- net defined by OOA [i] based on linear OOA(4931, 345, F49, 14, 14) (dual of [(345, 14), 4799, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4931, 2415, F49, 14) (dual of [2415, 2384, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(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(13) ⊂ Ce(8) [i] based on
- OA 7-folding and stacking [i] based on linear OA(4931, 2415, F49, 14) (dual of [2415, 2384, 15]-code), using
- net defined by OOA [i] based on linear OOA(4931, 345, F49, 14, 14) (dual of [(345, 14), 4799, 15]-NRT-code), using
(18, 18+14, 2485)-Net over F49 — Digital
Digital (18, 32, 2485)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4932, 2485, F49, 14) (dual of [2485, 2453, 15]-code), using
- 77 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 14 times 0, 1, 57 times 0) [i] based on linear OA(4927, 2403, F49, 14) (dual of [2403, 2376, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4925, 2401, F49, 13) (dual of [2401, 2376, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- 77 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 14 times 0, 1, 57 times 0) [i] based on linear OA(4927, 2403, F49, 14) (dual of [2403, 2376, 15]-code), using
(18, 18+14, 3752214)-Net in Base 49 — Upper bound on s
There is no (18, 32, 3752215)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 1 219760 824497 625949 010024 818712 071583 807314 184261 628465 > 4932 [i]