Best Known (138, 138+27, s)-Nets in Base 4
(138, 138+27, 5042)-Net over F4 — Constructive and digital
Digital (138, 165, 5042)-net over F4, using
- 43 times duplication [i] based on digital (135, 162, 5042)-net over F4, using
- net defined by OOA [i] based on linear OOA(4162, 5042, F4, 27, 27) (dual of [(5042, 27), 135972, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4162, 65547, F4, 27) (dual of [65547, 65385, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4162, 65554, F4, 27) (dual of [65554, 65392, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(4161, 65537, F4, 27) (dual of [65537, 65376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4145, 65537, F4, 25) (dual of [65537, 65392, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 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 C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4162, 65554, F4, 27) (dual of [65554, 65392, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4162, 65547, F4, 27) (dual of [65547, 65385, 28]-code), using
- net defined by OOA [i] based on linear OOA(4162, 5042, F4, 27, 27) (dual of [(5042, 27), 135972, 28]-NRT-code), using
(138, 138+27, 32779)-Net over F4 — Digital
Digital (138, 165, 32779)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4165, 32779, F4, 2, 27) (dual of [(32779, 2), 65393, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4165, 65558, F4, 27) (dual of [65558, 65393, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4145, 65536, F4, 25) (dual of [65536, 65391, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4137, 65536, F4, 23) (dual of [65536, 65399, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(41, 19, F4, 1) (dual of [19, 18, 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(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(4165, 65558, F4, 27) (dual of [65558, 65393, 28]-code), using
(138, 138+27, large)-Net in Base 4 — Upper bound on s
There is no (138, 165, large)-net in base 4, because
- 25 times m-reduction [i] would yield (138, 140, large)-net in base 4, but