Best Known (64, 64+16, s)-Nets in Base 4
(64, 64+16, 1062)-Net over F4 — Constructive and digital
Digital (64, 80, 1062)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (8, 16, 34)-net over F4, using
- trace code for nets [i] based on digital (0, 8, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- trace code for nets [i] based on digital (0, 8, 17)-net over F16, using
- digital (48, 64, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 16, 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, 16, 257)-net over F256, using
- digital (8, 16, 34)-net over F4, using
(64, 64+16, 4216)-Net over F4 — Digital
Digital (64, 80, 4216)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(480, 4216, F4, 16) (dual of [4216, 4136, 17]-code), using
- 101 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 26 times 0, 1, 52 times 0) [i] based on linear OA(474, 4109, F4, 16) (dual of [4109, 4035, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(473, 4096, F4, 17) (dual of [4096, 4023, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(461, 4096, F4, 14) (dual of [4096, 4035, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- 101 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 26 times 0, 1, 52 times 0) [i] based on linear OA(474, 4109, F4, 16) (dual of [4109, 4035, 17]-code), using
(64, 64+16, 1315729)-Net in Base 4 — Upper bound on s
There is no (64, 80, 1315730)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1 461504 024015 958564 069072 621259 119339 823348 408134 > 480 [i]