Best Known (34, 34+16, s)-Nets in Base 49
(34, 34+16, 14708)-Net over F49 — Constructive and digital
Digital (34, 50, 14708)-net over F49, using
- 491 times duplication [i] based on digital (33, 49, 14708)-net over F49, using
- net defined by OOA [i] based on linear OOA(4949, 14708, F49, 16, 16) (dual of [(14708, 16), 235279, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(4949, 117664, F49, 16) (dual of [117664, 117615, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [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(4934, 117649, F49, 12) (dual of [117649, 117615, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(493, 15, F49, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,49) or 15-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 Ce(15) ⊂ Ce(11) [i] based on
- OA 8-folding and stacking [i] based on linear OA(4949, 117664, F49, 16) (dual of [117664, 117615, 17]-code), using
- net defined by OOA [i] based on linear OOA(4949, 14708, F49, 16, 16) (dual of [(14708, 16), 235279, 17]-NRT-code), using
(34, 34+16, 103723)-Net over F49 — Digital
Digital (34, 50, 103723)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4950, 103723, F49, 16) (dual of [103723, 103673, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4950, 117668, F49, 16) (dual of [117668, 117618, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [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(4931, 117649, F49, 11) (dual of [117649, 117618, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(494, 19, F49, 4) (dual of [19, 15, 5]-code or 19-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(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(4950, 117668, F49, 16) (dual of [117668, 117618, 17]-code), using
(34, 34+16, large)-Net in Base 49 — Upper bound on s
There is no (34, 50, large)-net in base 49, because
- 14 times m-reduction [i] would yield (34, 36, large)-net in base 49, but