Best Known (26, 26+13, s)-Nets in Base 49
(26, 26+13, 19609)-Net over F49 — Constructive and digital
Digital (26, 39, 19609)-net over F49, using
- 491 times duplication [i] based on digital (25, 38, 19609)-net over F49, using
- net defined by OOA [i] based on linear OOA(4938, 19609, F49, 13, 13) (dual of [(19609, 13), 254879, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4938, 117655, F49, 13) (dual of [117655, 117617, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4931, 117650, F49, 11) (dual of [117650, 117619, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(491, 7, F49, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4938, 117655, F49, 13) (dual of [117655, 117617, 14]-code), using
- net defined by OOA [i] based on linear OOA(4938, 19609, F49, 13, 13) (dual of [(19609, 13), 254879, 14]-NRT-code), using
(26, 26+13, 70567)-Net over F49 — Digital
Digital (26, 39, 70567)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4939, 70567, F49, 13) (dual of [70567, 70528, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4939, 117660, F49, 13) (dual of [117660, 117621, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(4937, 117649, F49, 13) (dual of [117649, 117612, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(4928, 117649, F49, 10) (dual of [117649, 117621, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(492, 11, F49, 2) (dual of [11, 9, 3]-code or 11-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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(4939, 117660, F49, 13) (dual of [117660, 117621, 14]-code), using
(26, 26+13, large)-Net in Base 49 — Upper bound on s
There is no (26, 39, large)-net in base 49, because
- 11 times m-reduction [i] would yield (26, 28, large)-net in base 49, but