Best Known (246−44, 246, s)-Nets in Base 4
(246−44, 246, 1560)-Net over F4 — Constructive and digital
Digital (202, 246, 1560)-net over F4, using
- 41 times duplication [i] based on digital (201, 245, 1560)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 29, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (172, 216, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 72, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 72, 513)-net over F64, using
- digital (7, 29, 21)-net over F4, using
- (u, u+v)-construction [i] based on
(246−44, 246, 16441)-Net over F4 — Digital
Digital (202, 246, 16441)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4246, 16441, F4, 44) (dual of [16441, 16195, 45]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4245, 16439, F4, 44) (dual of [16439, 16194, 45]-code), using
- construction X applied to Ce(44) ⊂ Ce(36) [i] based on
- linear OA(4232, 16384, F4, 45) (dual of [16384, 16152, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(4190, 16384, F4, 37) (dual of [16384, 16194, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(413, 55, F4, 6) (dual of [55, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(44) ⊂ Ce(36) [i] based on
- linear OA(4245, 16440, F4, 43) (dual of [16440, 16195, 44]-code), using Gilbert–Varšamov bound and bm = 4245 > Vbs−1(k−1) = 86 255439 833587 769115 546280 187470 614841 074447 503624 857998 096348 267507 577990 581222 545492 634860 194345 453600 888024 467279 216903 315092 704224 773718 491482 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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
- linear OA(4245, 16439, F4, 44) (dual of [16439, 16194, 45]-code), using
- construction X with Varšamov bound [i] based on
(246−44, 246, large)-Net in Base 4 — Upper bound on s
There is no (202, 246, large)-net in base 4, because
- 42 times m-reduction [i] would yield (202, 204, large)-net in base 4, but