Best Known (138, 138+39, s)-Nets in Base 4
(138, 138+39, 1048)-Net over F4 — Constructive and digital
Digital (138, 177, 1048)-net over F4, using
- 41 times duplication [i] based on digital (137, 176, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 44, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
- trace code for nets [i] based on digital (5, 44, 262)-net over F256, using
(138, 138+39, 3541)-Net over F4 — Digital
Digital (138, 177, 3541)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4177, 3541, F4, 39) (dual of [3541, 3364, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4177, 4110, F4, 39) (dual of [4110, 3933, 40]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4176, 4109, F4, 39) (dual of [4109, 3933, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(36) [i] based on
- 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(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(38) ⊂ Ce(36) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4176, 4109, F4, 39) (dual of [4109, 3933, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4177, 4110, F4, 39) (dual of [4110, 3933, 40]-code), using
(138, 138+39, 997847)-Net in Base 4 — Upper bound on s
There is no (138, 177, 997848)-net in base 4, because
- 1 times m-reduction [i] would yield (138, 176, 997848)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 9174 015019 798001 301148 471140 660968 055589 358992 801476 602404 930885 072398 044690 195230 294383 455779 475605 691581 > 4176 [i]