Best Known (53, 53+20, s)-Nets in Base 4
(53, 53+20, 312)-Net over F4 — Constructive and digital
Digital (53, 73, 312)-net over F4, using
- 2 times m-reduction [i] based on digital (53, 75, 312)-net over F4, using
- trace code for nets [i] based on digital (3, 25, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 25, 104)-net over F64, using
(53, 53+20, 387)-Net in Base 4 — Constructive
(53, 73, 387)-net in base 4, using
- 41 times duplication [i] based on (52, 72, 387)-net in base 4, using
- trace code for nets [i] based on (4, 24, 129)-net in base 64, using
- 4 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
- base change [i] based on digital (0, 24, 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, 24, 129)-net over F128, using
- 4 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
- trace code for nets [i] based on (4, 24, 129)-net in base 64, using
(53, 53+20, 554)-Net over F4 — Digital
Digital (53, 73, 554)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(473, 554, F4, 20) (dual of [554, 481, 21]-code), using
- 35 step Varšamov–Edel lengthening with (ri) = (1, 34 times 0) [i] based on linear OA(472, 518, F4, 20) (dual of [518, 446, 21]-code), using
- trace code [i] based on linear OA(1636, 259, F16, 20) (dual of [259, 223, 21]-code), using
- construction XX applied to C1 = C([254,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([254,18]) [i] based on
- linear OA(1634, 255, F16, 19) (dual of [255, 221, 20]-code), using the primitive BCH-code C(I) with length 255 = 162−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1634, 255, F16, 19) (dual of [255, 221, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 162−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1636, 255, F16, 20) (dual of [255, 219, 21]-code), using the primitive BCH-code C(I) with length 255 = 162−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1632, 255, F16, 18) (dual of [255, 223, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 162−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(160, 2, F16, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(160, 2, F16, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([254,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([254,18]) [i] based on
- trace code [i] based on linear OA(1636, 259, F16, 20) (dual of [259, 223, 21]-code), using
- 35 step Varšamov–Edel lengthening with (ri) = (1, 34 times 0) [i] based on linear OA(472, 518, F4, 20) (dual of [518, 446, 21]-code), using
(53, 53+20, 37480)-Net in Base 4 — Upper bound on s
There is no (53, 73, 37481)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 89 221329 822931 128228 046186 406260 993795 156960 > 473 [i]