Best Known (42, 53, s)-Nets in Base 4
(42, 53, 1064)-Net over F4 — Constructive and digital
Digital (42, 53, 1064)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 9, 36)-net over F4, using
- digital (33, 44, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 11, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 11, 257)-net over F256, using
(42, 53, 4133)-Net over F4 — Digital
Digital (42, 53, 4133)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(453, 4133, F4, 11) (dual of [4133, 4080, 12]-code), using
- 27 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 16 times 0) [i] based on linear OA(449, 4102, F4, 11) (dual of [4102, 4053, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(449, 4096, F4, 11) (dual of [4096, 4047, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(443, 4096, F4, 10) (dual of [4096, 4053, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 27 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 16 times 0) [i] based on linear OA(449, 4102, F4, 11) (dual of [4102, 4053, 12]-code), using
(42, 53, 1585396)-Net in Base 4 — Upper bound on s
There is no (42, 53, 1585397)-net in base 4, because
- 1 times m-reduction [i] would yield (42, 52, 1585397)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 20 282440 468588 104333 942670 124596 > 452 [i]