Best Known (112, 134, s)-Nets in Base 5
(112, 134, 7118)-Net over F5 — Constructive and digital
Digital (112, 134, 7118)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 16)-net over F5, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 3 and N(F) ≥ 16, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- digital (98, 120, 7102)-net over F5, using
- net defined by OOA [i] based on linear OOA(5120, 7102, F5, 22, 22) (dual of [(7102, 22), 156124, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5120, 78122, F5, 22) (dual of [78122, 78002, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(5120, 78122, F5, 22) (dual of [78122, 78002, 23]-code), using
- net defined by OOA [i] based on linear OOA(5120, 7102, F5, 22, 22) (dual of [(7102, 22), 156124, 23]-NRT-code), using
- digital (3, 14, 16)-net over F5, using
(112, 134, 78182)-Net over F5 — Digital
Digital (112, 134, 78182)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5134, 78182, F5, 22) (dual of [78182, 78048, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5133, 78180, F5, 22) (dual of [78180, 78047, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(13) [i] based on
- linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(513, 55, F5, 7) (dual of [55, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(513, 63, F5, 7) (dual of [63, 50, 8]-code), using
- a “GraCyc†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(513, 63, F5, 7) (dual of [63, 50, 8]-code), using
- construction X applied to Ce(21) ⊂ Ce(13) [i] based on
- linear OA(5133, 78181, F5, 21) (dual of [78181, 78048, 22]-code), using Gilbert–Varšamov bound and bm = 5133 > Vbs−1(k−1) = 32 806565 410812 207250 510845 967978 929410 968263 921040 994583 886449 120376 063787 004305 770474 169585 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(5133, 78180, F5, 22) (dual of [78180, 78047, 23]-code), using
- construction X with Varšamov bound [i] based on
(112, 134, large)-Net in Base 5 — Upper bound on s
There is no (112, 134, large)-net in base 5, because
- 20 times m-reduction [i] would yield (112, 114, large)-net in base 5, but