Best Known (27, 27+26, s)-Nets in Base 49
(27, 27+26, 344)-Net over F49 — Constructive and digital
Digital (27, 53, 344)-net over F49, using
- t-expansion [i] based on digital (21, 53, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
(27, 27+26, 1088)-Net over F49 — Digital
Digital (27, 53, 1088)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4953, 1088, F49, 2, 26) (dual of [(1088, 2), 2123, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4953, 1204, F49, 2, 26) (dual of [(1204, 2), 2355, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4953, 2408, F49, 26) (dual of [2408, 2355, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4953, 2409, F49, 26) (dual of [2409, 2356, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(4951, 2401, F49, 26) (dual of [2401, 2350, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4945, 2401, F49, 23) (dual of [2401, 2356, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(492, 8, F49, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,49)), using
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- Reed–Solomon code RS(47,49) [i]
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4953, 2409, F49, 26) (dual of [2409, 2356, 27]-code), using
- OOA 2-folding [i] based on linear OA(4953, 2408, F49, 26) (dual of [2408, 2355, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(4953, 1204, F49, 2, 26) (dual of [(1204, 2), 2355, 27]-NRT-code), using
(27, 27+26, 918250)-Net in Base 49 — Upper bound on s
There is no (27, 53, 918251)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 380533 289139 951918 429085 424977 620494 220759 956640 220186 603133 780867 867931 176460 790726 768849 > 4953 [i]