Best Known (148−24, 148, s)-Nets in Base 5
(148−24, 148, 6523)-Net over F5 — Constructive and digital
Digital (124, 148, 6523)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (110, 134, 6511)-net over F5, using
- net defined by OOA [i] based on linear OOA(5134, 6511, F5, 24, 24) (dual of [(6511, 24), 156130, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(5134, 78132, F5, 24) (dual of [78132, 77998, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(5134, 78125, F5, 24) (dual of [78125, 77991, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(5127, 78125, F5, 23) (dual of [78125, 77998, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 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
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- OA 12-folding and stacking [i] based on linear OA(5134, 78132, F5, 24) (dual of [78132, 77998, 25]-code), using
- net defined by OOA [i] based on linear OOA(5134, 6511, F5, 24, 24) (dual of [(6511, 24), 156130, 25]-NRT-code), using
- digital (2, 14, 12)-net over F5, using
(148−24, 148, 78189)-Net over F5 — Digital
Digital (124, 148, 78189)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5148, 78189, F5, 24) (dual of [78189, 78041, 25]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5147, 78187, F5, 24) (dual of [78187, 78040, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- linear OA(5134, 78125, F5, 24) (dual of [78125, 77991, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(513, 62, F5, 7) (dual of [62, 49, 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(23) ⊂ Ce(15) [i] based on
- linear OA(5147, 78188, F5, 23) (dual of [78188, 78041, 24]-code), using Gilbert–Varšamov bound and bm = 5147 > Vbs−1(k−1) = 6954 415740 943782 753630 051600 992062 632249 132417 623138 635593 651920 378621 944793 628392 308548 326966 386525 [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(5147, 78187, F5, 24) (dual of [78187, 78040, 25]-code), using
- construction X with Varšamov bound [i] based on
(148−24, 148, large)-Net in Base 5 — Upper bound on s
There is no (124, 148, large)-net in base 5, because
- 22 times m-reduction [i] would yield (124, 126, large)-net in base 5, but