Best Known (178−24, 178, s)-Nets in Base 4
(178−24, 178, 21860)-Net over F4 — Constructive and digital
Digital (154, 178, 21860)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 16, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- digital (138, 162, 21845)-net over F4, using
- net defined by OOA [i] based on linear OOA(4162, 21845, F4, 24, 24) (dual of [(21845, 24), 524118, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4162, 262140, F4, 24) (dual of [262140, 261978, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4162, 262144, F4, 24) (dual of [262144, 261982, 25]-code), using
- 1 times truncation [i] based on linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4162, 262144, F4, 24) (dual of [262144, 261982, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(4162, 262140, F4, 24) (dual of [262140, 261978, 25]-code), using
- net defined by OOA [i] based on linear OOA(4162, 21845, F4, 24, 24) (dual of [(21845, 24), 524118, 25]-NRT-code), using
- digital (4, 16, 15)-net over F4, using
(178−24, 178, 210641)-Net over F4 — Digital
Digital (154, 178, 210641)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4178, 210641, F4, 24) (dual of [210641, 210463, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4178, 262214, F4, 24) (dual of [262214, 262036, 25]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4176, 262212, F4, 24) (dual of [262212, 262036, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4109, 262145, F4, 17) (dual of [262145, 262036, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(413, 67, F4, 6) (dual of [67, 54, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(413, 64, F4, 6) (dual of [64, 51, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 3, F4, 0) (dual of [3, 3, 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
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4176, 262212, F4, 24) (dual of [262212, 262036, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4178, 262214, F4, 24) (dual of [262214, 262036, 25]-code), using
(178−24, 178, large)-Net in Base 4 — Upper bound on s
There is no (154, 178, large)-net in base 4, because
- 22 times m-reduction [i] would yield (154, 156, large)-net in base 4, but