Best Known (48−16, 48, s)-Nets in Base 49
(48−16, 48, 14707)-Net over F49 — Constructive and digital
Digital (32, 48, 14707)-net over F49, using
- 491 times duplication [i] based on digital (31, 47, 14707)-net over F49, using
- net defined by OOA [i] based on linear OOA(4947, 14707, F49, 16, 16) (dual of [(14707, 16), 235265, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(4947, 117656, F49, 16) (dual of [117656, 117609, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(4946, 117649, F49, 16) (dual of [117649, 117603, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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 Ce(15) ⊂ Ce(13) [i] based on
- OA 8-folding and stacking [i] based on linear OA(4947, 117656, F49, 16) (dual of [117656, 117609, 17]-code), using
- net defined by OOA [i] based on linear OOA(4947, 14707, F49, 16, 16) (dual of [(14707, 16), 235265, 17]-NRT-code), using
(48−16, 48, 59484)-Net over F49 — Digital
Digital (32, 48, 59484)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4948, 59484, F49, 16) (dual of [59484, 59436, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4948, 117660, F49, 16) (dual of [117660, 117612, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(4946, 117649, F49, 16) (dual of [117649, 117603, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 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(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(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4948, 117660, F49, 16) (dual of [117660, 117612, 17]-code), using
(48−16, 48, large)-Net in Base 49 — Upper bound on s
There is no (32, 48, large)-net in base 49, because
- 14 times m-reduction [i] would yield (32, 34, large)-net in base 49, but