Best Known (138−23, 138, s)-Nets in Base 5
(138−23, 138, 7108)-Net over F5 — Constructive and digital
Digital (115, 138, 7108)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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 (104, 127, 7102)-net over F5, using
- net defined by OOA [i] based on linear OOA(5127, 7102, F5, 23, 23) (dual of [(7102, 23), 163219, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5127, 78123, F5, 23) (dual of [78123, 77996, 24]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(5127, 78125, F5, 23) (dual of [78125, 77998, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5127, 78123, F5, 23) (dual of [78123, 77996, 24]-code), using
- net defined by OOA [i] based on linear OOA(5127, 7102, F5, 23, 23) (dual of [(7102, 23), 163219, 24]-NRT-code), using
- digital (0, 11, 6)-net over F5, using
(138−23, 138, 78172)-Net over F5 — Digital
Digital (115, 138, 78172)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5138, 78172, F5, 23) (dual of [78172, 78034, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5136, 78169, F5, 23) (dual of [78169, 78033, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- 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(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(59, 44, F5, 5) (dual of [44, 35, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(5136, 78170, F5, 21) (dual of [78170, 78034, 22]-code), using Gilbert–Varšamov bound and bm = 5136 > Vbs−1(k−1) = 32 714359 513332 999726 313542 034775 151021 407241 742771 897852 108779 094423 408600 157540 183212 247845 [i]
- linear OA(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(5136, 78169, F5, 23) (dual of [78169, 78033, 24]-code), using
- construction X with Varšamov bound [i] based on
(138−23, 138, large)-Net in Base 5 — Upper bound on s
There is no (115, 138, large)-net in base 5, because
- 21 times m-reduction [i] would yield (115, 117, large)-net in base 5, but