Best Known (147−24, 147, s)-Nets in Base 4
(147−24, 147, 5462)-Net over F4 — Constructive and digital
Digital (123, 147, 5462)-net over F4, using
- 41 times duplication [i] based on digital (122, 146, 5462)-net over F4, using
- t-expansion [i] based on digital (121, 146, 5462)-net over F4, using
- net defined by OOA [i] based on linear OOA(4146, 5462, F4, 25, 25) (dual of [(5462, 25), 136404, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4146, 65545, F4, 25) (dual of [65545, 65399, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 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, 9, F4, 1) (dual of [9, 8, 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(24) ⊂ Ce(22) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(4146, 65545, F4, 25) (dual of [65545, 65399, 26]-code), using
- net defined by OOA [i] based on linear OOA(4146, 5462, F4, 25, 25) (dual of [(5462, 25), 136404, 26]-NRT-code), using
- t-expansion [i] based on digital (121, 146, 5462)-net over F4, using
(147−24, 147, 32777)-Net over F4 — Digital
Digital (123, 147, 32777)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4147, 32777, F4, 2, 24) (dual of [(32777, 2), 65407, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4147, 65554, F4, 24) (dual of [65554, 65407, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4147, 65555, F4, 24) (dual of [65555, 65408, 25]-code), using
- construction XX applied to Ce(24) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- 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(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(41, 18, F4, 1) (dual of [18, 17, 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(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(24) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4147, 65555, F4, 24) (dual of [65555, 65408, 25]-code), using
- OOA 2-folding [i] based on linear OA(4147, 65554, F4, 24) (dual of [65554, 65407, 25]-code), using
(147−24, 147, large)-Net in Base 4 — Upper bound on s
There is no (123, 147, large)-net in base 4, because
- 22 times m-reduction [i] would yield (123, 125, large)-net in base 4, but