Best Known (67, 93, s)-Nets in Base 7
(67, 93, 344)-Net over F7 — Constructive and digital
Digital (67, 93, 344)-net over F7, using
- base reduction for projective spaces (embedding PG(46,49) in PG(92,7)) for nets [i] based on digital (21, 47, 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
(67, 93, 2495)-Net over F7 — Digital
Digital (67, 93, 2495)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(793, 2495, F7, 26) (dual of [2495, 2402, 27]-code), using
- 86 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 21 times 0, 1, 58 times 0) [i] based on linear OA(789, 2405, F7, 26) (dual of [2405, 2316, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(789, 2401, F7, 26) (dual of [2401, 2312, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(785, 2401, F7, 25) (dual of [2401, 2316, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(25) ⊂ Ce(24) [i] based on
- 86 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 21 times 0, 1, 58 times 0) [i] based on linear OA(789, 2405, F7, 26) (dual of [2405, 2316, 27]-code), using
(67, 93, 1049428)-Net in Base 7 — Upper bound on s
There is no (67, 93, 1049429)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 3 927531 453993 994717 731212 009836 947162 970697 823764 465302 352412 021716 892837 902527 > 793 [i]