Best Known (197−43, 197, s)-Nets in Base 4
(197−43, 197, 1052)-Net over F4 — Constructive and digital
Digital (154, 197, 1052)-net over F4, using
- 41 times duplication [i] based on digital (153, 196, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 49, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
- trace code for nets [i] based on digital (6, 49, 263)-net over F256, using
(197−43, 197, 4031)-Net over F4 — Digital
Digital (154, 197, 4031)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4197, 4031, F4, 43) (dual of [4031, 3834, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4197, 4114, F4, 43) (dual of [4114, 3917, 44]-code), using
- construction XX applied to Ce(42) ⊂ Ce(40) ⊂ Ce(38) [i] based on
- linear OA(4193, 4096, F4, 43) (dual of [4096, 3903, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(4181, 4096, F4, 41) (dual of [4096, 3915, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4175, 4096, F4, 39) (dual of [4096, 3921, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 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(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(42) ⊂ Ce(40) ⊂ Ce(38) [i] based on
- discarding factors / shortening the dual code based on linear OA(4197, 4114, F4, 43) (dual of [4114, 3917, 44]-code), using
(197−43, 197, 1203903)-Net in Base 4 — Upper bound on s
There is no (154, 197, 1203904)-net in base 4, because
- 1 times m-reduction [i] would yield (154, 196, 1203904)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 10086 958292 805596 215795 407882 277286 130596 843469 199601 362907 565123 697023 663778 971518 521078 401447 174426 574376 703044 976973 > 4196 [i]