Best Known (138, 138+37, s)-Nets in Base 4
(138, 138+37, 1052)-Net over F4 — Constructive and digital
Digital (138, 175, 1052)-net over F4, using
- 1 times m-reduction [i] based on digital (138, 176, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 44, 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, 44, 263)-net over F256, using
(138, 138+37, 4259)-Net over F4 — Digital
Digital (138, 175, 4259)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4175, 4259, F4, 37) (dual of [4259, 4084, 38]-code), using
- 145 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0, 1, 22 times 0, 1, 34 times 0, 1, 52 times 0) [i] based on linear OA(4164, 4103, F4, 37) (dual of [4103, 3939, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- 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(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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(36) ⊂ Ce(34) [i] based on
- 145 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0, 1, 22 times 0, 1, 34 times 0, 1, 52 times 0) [i] based on linear OA(4164, 4103, F4, 37) (dual of [4103, 3939, 38]-code), using
(138, 138+37, 1663099)-Net in Base 4 — Upper bound on s
There is no (138, 175, 1663100)-net in base 4, because
- 1 times m-reduction [i] would yield (138, 174, 1663100)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 573 376995 439957 395434 581743 114658 796607 423336 484961 550254 779051 109316 942896 240771 314271 185727 565093 517531 > 4174 [i]