Best Known (118, 141, s)-Nets in Base 5
(118, 141, 7118)-Net over F5 — Constructive and digital
Digital (118, 141, 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 (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 (3, 14, 16)-net over F5, using
(118, 141, 78183)-Net over F5 — Digital
Digital (118, 141, 78183)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5141, 78183, F5, 23) (dual of [78183, 78042, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5139, 78179, F5, 23) (dual of [78179, 78040, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(15) [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(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(512, 54, F5, 6) (dual of [54, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(512, 62, F5, 6) (dual of [62, 50, 7]-code), using
- the cyclic code C(A) with length 62 | 53−1, defining set A = {4,8,11,17}, and minimum distance d ≥ |{8,11,14,…,23}|+1 = 7 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(512, 62, F5, 6) (dual of [62, 50, 7]-code), using
- construction X applied to Ce(22) ⊂ Ce(15) [i] based on
- linear OA(5139, 78181, F5, 22) (dual of [78181, 78042, 23]-code), using Gilbert–Varšamov bound and bm = 5139 > Vbs−1(k−1) = 488413 210814 076622 369628 964373 928239 773280 549482 200353 502881 535226 617139 321378 809333 547408 876785 [i]
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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(5139, 78179, F5, 23) (dual of [78179, 78040, 24]-code), using
- construction X with Varšamov bound [i] based on
(118, 141, large)-Net in Base 5 — Upper bound on s
There is no (118, 141, large)-net in base 5, because
- 21 times m-reduction [i] would yield (118, 120, large)-net in base 5, but