Best Known (117, 117+23, s)-Nets in Base 5
(117, 117+23, 7114)-Net over F5 — Constructive and digital
Digital (117, 140, 7114)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 13, 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 (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 (2, 13, 12)-net over F5, using
(117, 117+23, 78181)-Net over F5 — Digital
Digital (117, 140, 78181)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5140, 78181, F5, 23) (dual of [78181, 78041, 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, 78180, F5, 22) (dual of [78180, 78041, 23]-code), using Gilbert–Varšamov bound and bm = 5139 > Vbs−1(k−1) = 488282 018122 615488 776625 588723 256827 563722 393840 563813 717573 349059 141483 832043 660451 166042 939645 [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(5139, 78179, F5, 23) (dual of [78179, 78040, 24]-code), using
- construction X with Varšamov bound [i] based on
(117, 117+23, large)-Net in Base 5 — Upper bound on s
There is no (117, 140, large)-net in base 5, because
- 21 times m-reduction [i] would yield (117, 119, large)-net in base 5, but