Best Known (77, 77+32, s)-Nets in Base 7
(77, 77+32, 688)-Net over F7 — Constructive and digital
Digital (77, 109, 688)-net over F7, using
- t-expansion [i] based on digital (76, 109, 688)-net over F7, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on digital (76, 110, 688)-net over F7, using
(77, 77+32, 2195)-Net over F7 — Digital
Digital (77, 109, 2195)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7109, 2195, F7, 32) (dual of [2195, 2086, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(7109, 2401, F7, 32) (dual of [2401, 2292, 33]-code), using
- an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- discarding factors / shortening the dual code based on linear OA(7109, 2401, F7, 32) (dual of [2401, 2292, 33]-code), using
(77, 77+32, 648053)-Net in Base 7 — Upper bound on s
There is no (77, 109, 648054)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 130 523452 056507 161659 967696 653787 075291 326945 148539 781454 301323 706312 454432 465334 616417 990945 > 7109 [i]