Best Known (54−9, 54, s)-Nets in Base 5
(54−9, 54, 19538)-Net over F5 — Constructive and digital
Digital (45, 54, 19538)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (41, 50, 19532)-net over F5, using
- net defined by OOA [i] based on linear OOA(550, 19532, F5, 9, 9) (dual of [(19532, 9), 175738, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(550, 78129, F5, 9) (dual of [78129, 78079, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(550, 78132, F5, 9) (dual of [78132, 78082, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(550, 78125, F5, 9) (dual of [78125, 78075, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(550, 78132, F5, 9) (dual of [78132, 78082, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(550, 78129, F5, 9) (dual of [78129, 78079, 10]-code), using
- net defined by OOA [i] based on linear OOA(550, 19532, F5, 9, 9) (dual of [(19532, 9), 175738, 10]-NRT-code), using
- digital (0, 4, 6)-net over F5, using
(54−9, 54, 78151)-Net over F5 — Digital
Digital (45, 54, 78151)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(554, 78151, F5, 9) (dual of [78151, 78097, 10]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(553, 78149, F5, 9) (dual of [78149, 78096, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(550, 78125, F5, 9) (dual of [78125, 78075, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(553, 78150, F5, 8) (dual of [78150, 78097, 9]-code), using Gilbert–Varšamov bound and bm = 553 > Vbs−1(k−1) = 57855 732842 240392 377082 162564 064053 [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(553, 78149, F5, 9) (dual of [78149, 78096, 10]-code), using
- construction X with Varšamov bound [i] based on
(54−9, 54, large)-Net in Base 5 — Upper bound on s
There is no (45, 54, large)-net in base 5, because
- 7 times m-reduction [i] would yield (45, 47, large)-net in base 5, but