Best Known (79, 110, s)-Nets in Base 7
(79, 110, 688)-Net over F7 — Constructive and digital
Digital (79, 110, 688)-net over F7, using
- t-expansion [i] based on digital (76, 110, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 55, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- trace code for nets [i] based on digital (21, 55, 344)-net over F49, using
(79, 110, 2574)-Net over F7 — Digital
Digital (79, 110, 2574)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7110, 2574, F7, 31) (dual of [2574, 2464, 32]-code), using
- 164 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 17 times 0, 1, 47 times 0, 1, 92 times 0) [i] based on linear OA(7105, 2405, F7, 31) (dual of [2405, 2300, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(7105, 2401, F7, 31) (dual of [2401, 2296, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(7101, 2401, F7, 30) (dual of [2401, 2300, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- 164 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 17 times 0, 1, 47 times 0, 1, 92 times 0) [i] based on linear OA(7105, 2405, F7, 31) (dual of [2405, 2300, 32]-code), using
(79, 110, 1481352)-Net in Base 7 — Upper bound on s
There is no (79, 110, 1481353)-net in base 7, because
- 1 times m-reduction [i] would yield (79, 109, 1481353)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 130 523689 645968 736638 730107 074875 036723 086563 632444 370483 738505 967347 050040 860123 305603 370743 > 7109 [i]