Best Known (63, 63+17, s)-Nets in Base 4
(63, 63+17, 1045)-Net over F4 — Constructive and digital
Digital (63, 80, 1045)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 17)-net over F4, using
- digital (51, 68, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 17, 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, 17, 257)-net over F256, using
(63, 63+17, 3162)-Net over F4 — Digital
Digital (63, 80, 3162)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(480, 3162, F4, 17) (dual of [3162, 3082, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(480, 4122, F4, 17) (dual of [4122, 4042, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(10) [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(455, 4096, F4, 13) (dual of [4096, 4041, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(45, 24, F4, 3) (dual of [24, 19, 4]-code or 24-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(480, 4122, F4, 17) (dual of [4122, 4042, 18]-code), using
(63, 63+17, 1106391)-Net in Base 4 — Upper bound on s
There is no (63, 80, 1106392)-net in base 4, because
- 1 times m-reduction [i] would yield (63, 79, 1106392)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 365376 906086 645514 695085 931885 623389 523706 364194 > 479 [i]