Best Known (15, 15+6, s)-Nets in Base 4
(15, 15+6, 343)-Net over F4 — Constructive and digital
Digital (15, 21, 343)-net over F4, using
- net defined by OOA [i] based on linear OOA(421, 343, F4, 6, 6) (dual of [(343, 6), 2037, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(421, 1029, F4, 6) (dual of [1029, 1008, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(421, 1024, F4, 6) (dual of [1024, 1003, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(416, 1024, F4, 5) (dual of [1024, 1008, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(421, 1029, F4, 6) (dual of [1029, 1008, 7]-code), using
(15, 15+6, 387)-Net in Base 4 — Constructive
(15, 21, 387)-net in base 4, using
- trace code for nets [i] based on (1, 7, 129)-net in base 64, using
- base change [i] based on digital (0, 6, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 6, 129)-net over F128, using
(15, 15+6, 753)-Net over F4 — Digital
Digital (15, 21, 753)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(421, 753, F4, 6) (dual of [753, 732, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(421, 1023, F4, 6) (dual of [1023, 1002, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(421, 1023, F4, 6) (dual of [1023, 1002, 7]-code), using
(15, 15+6, 9921)-Net in Base 4 — Upper bound on s
There is no (15, 21, 9922)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 4 398177 561328 > 421 [i]