Best Known (79, 110, s)-Nets in Base 9
(79, 110, 756)-Net over F9 — Constructive and digital
Digital (79, 110, 756)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 16)-net over F9, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 1 and N(F) ≥ 16, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- digital (63, 94, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 47, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 47, 370)-net over F81, using
- digital (1, 16, 16)-net over F9, using
(79, 110, 5615)-Net over F9 — Digital
Digital (79, 110, 5615)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9110, 5615, F9, 31) (dual of [5615, 5505, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(9110, 6570, F9, 31) (dual of [6570, 6460, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(9109, 6561, F9, 31) (dual of [6561, 6452, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(9101, 6561, F9, 29) (dual of [6561, 6460, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(9110, 6570, F9, 31) (dual of [6570, 6460, 32]-code), using
(79, 110, 6899829)-Net in Base 9 — Upper bound on s
There is no (79, 110, 6899830)-net in base 9, because
- 1 times m-reduction [i] would yield (79, 109, 6899830)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 102 904515 433372 943224 080776 147748 103069 648988 369146 665951 781275 654993 755651 473180 077479 070491 515522 970769 > 9109 [i]