Best Known (60, 60+22, s)-Nets in Base 4
(60, 60+22, 384)-Net over F4 — Constructive and digital
Digital (60, 82, 384)-net over F4, using
- 41 times duplication [i] based on digital (59, 81, 384)-net over F4, using
- trace code for nets [i] based on digital (5, 27, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 27, 128)-net over F64, using
(60, 60+22, 450)-Net in Base 4 — Constructive
(60, 82, 450)-net in base 4, using
- 41 times duplication [i] based on (59, 81, 450)-net in base 4, using
- trace code for nets [i] based on (5, 27, 150)-net in base 64, using
- 1 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
- base change [i] based on digital (1, 24, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 24, 150)-net over F128, using
- 1 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
- trace code for nets [i] based on (5, 27, 150)-net in base 64, using
(60, 60+22, 745)-Net over F4 — Digital
Digital (60, 82, 745)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(482, 745, F4, 22) (dual of [745, 663, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(482, 1034, F4, 22) (dual of [1034, 952, 23]-code), using
- construction XX applied to C1 = C([321,341]), C2 = C([323,342]), C3 = C1 + C2 = C([323,341]), and C∩ = C1 ∩ C2 = C([321,342]) [i] based on
- linear OA(476, 1023, F4, 21) (dual of [1023, 947, 22]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {321,322,…,341}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(476, 1023, F4, 20) (dual of [1023, 947, 21]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {323,324,…,342}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(481, 1023, F4, 22) (dual of [1023, 942, 23]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {321,322,…,342}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(471, 1023, F4, 19) (dual of [1023, 952, 20]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {323,324,…,341}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(41, 6, F4, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- 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 XX applied to C1 = C([321,341]), C2 = C([323,342]), C3 = C1 + C2 = C([323,341]), and C∩ = C1 ∩ C2 = C([321,342]) [i] based on
- discarding factors / shortening the dual code based on linear OA(482, 1034, F4, 22) (dual of [1034, 952, 23]-code), using
(60, 60+22, 50338)-Net in Base 4 — Upper bound on s
There is no (60, 82, 50339)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 23 385726 107207 625176 316471 437235 609262 876422 917104 > 482 [i]