Best Known (28, 28+16, s)-Nets in Base 7
(28, 28+16, 116)-Net over F7 — Constructive and digital
Digital (28, 44, 116)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 16)-net over F7, using
- 4 times m-reduction [i] based on digital (4, 16, 16)-net over F7, using
- digital (16, 32, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 16, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- trace code for nets [i] based on digital (0, 16, 50)-net over F49, using
- digital (4, 12, 16)-net over F7, using
(28, 28+16, 367)-Net over F7 — Digital
Digital (28, 44, 367)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(744, 367, F7, 16) (dual of [367, 323, 17]-code), using
- 17 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 10 times 0) [i] based on linear OA(740, 346, F7, 16) (dual of [346, 306, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(740, 343, F7, 16) (dual of [343, 303, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(737, 343, F7, 15) (dual of [343, 306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 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(15) ⊂ Ce(14) [i] based on
- 17 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 10 times 0) [i] based on linear OA(740, 346, F7, 16) (dual of [346, 306, 17]-code), using
(28, 28+16, 27893)-Net in Base 7 — Upper bound on s
There is no (28, 44, 27894)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 15 288954 319373 486634 111379 474909 952785 > 744 [i]