Best Known (11, 20, s)-Nets in Base 49
(11, 20, 603)-Net over F49 — Constructive and digital
Digital (11, 20, 603)-net over F49, using
- net defined by OOA [i] based on linear OOA(4920, 603, F49, 9, 9) (dual of [(603, 9), 5407, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4920, 2413, F49, 9) (dual of [2413, 2393, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(4917, 2402, F49, 9) (dual of [2402, 2385, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(499, 2402, F49, 5) (dual of [2402, 2393, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(493, 11, F49, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,49) or 11-cap in PG(2,49)), using
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- Reed–Solomon code RS(46,49) [i]
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(4920, 2413, F49, 9) (dual of [2413, 2393, 10]-code), using
(11, 20, 2444)-Net over F49 — Digital
Digital (11, 20, 2444)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4920, 2444, F49, 9) (dual of [2444, 2424, 10]-code), using
- 38 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 32 times 0) [i] based on linear OA(4917, 2403, F49, 9) (dual of [2403, 2386, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- 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(4915, 2401, F49, 8) (dual of [2401, 2386, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- 38 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 32 times 0) [i] based on linear OA(4917, 2403, F49, 9) (dual of [2403, 2386, 10]-code), using
(11, 20, 4923147)-Net in Base 49 — Upper bound on s
There is no (11, 20, 4923148)-net in base 49, because
- 1 times m-reduction [i] would yield (11, 19, 4923148)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 129 934905 085185 361116 265902 685441 > 4919 [i]