Best Known (127, 127+24, s)-Nets in Base 4
(127, 127+24, 5463)-Net over F4 — Constructive and digital
Digital (127, 151, 5463)-net over F4, using
- 42 times duplication [i] based on digital (125, 149, 5463)-net over F4, using
- t-expansion [i] based on digital (124, 149, 5463)-net over F4, using
- net defined by OOA [i] based on linear OOA(4149, 5463, F4, 25, 25) (dual of [(5463, 25), 136426, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4149, 65557, F4, 25) (dual of [65557, 65408, 26]-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(43, 20, F4, 2) (dual of [20, 17, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), 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
- OOA 12-folding and stacking with additional row [i] based on linear OA(4149, 65557, F4, 25) (dual of [65557, 65408, 26]-code), using
- net defined by OOA [i] based on linear OOA(4149, 5463, F4, 25, 25) (dual of [(5463, 25), 136426, 26]-NRT-code), using
- t-expansion [i] based on digital (124, 149, 5463)-net over F4, using
(127, 127+24, 38414)-Net over F4 — Digital
Digital (127, 151, 38414)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4151, 38414, F4, 24) (dual of [38414, 38263, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4151, 65567, F4, 24) (dual of [65567, 65416, 25]-code), using
- 1 times truncation [i] based on linear OA(4152, 65568, F4, 25) (dual of [65568, 65416, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(20) ⊂ Ce(18) [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(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(4113, 65536, F4, 19) (dual of [65536, 65423, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(45, 30, F4, 3) (dual of [30, 25, 4]-code or 30-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(24) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- 1 times truncation [i] based on linear OA(4152, 65568, F4, 25) (dual of [65568, 65416, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4151, 65567, F4, 24) (dual of [65567, 65416, 25]-code), using
(127, 127+24, large)-Net in Base 4 — Upper bound on s
There is no (127, 151, large)-net in base 4, because
- 22 times m-reduction [i] would yield (127, 129, large)-net in base 4, but