Best Known (181, 181+31, s)-Nets in Base 4
(181, 181+31, 17477)-Net over F4 — Constructive and digital
Digital (181, 212, 17477)-net over F4, using
- 43 times duplication [i] based on digital (178, 209, 17477)-net over F4, using
- net defined by OOA [i] based on linear OOA(4209, 17477, F4, 31, 31) (dual of [(17477, 31), 541578, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4209, 262156, F4, 31) (dual of [262156, 261947, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, 262163, F4, 31) (dual of [262163, 261954, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4209, 262163, F4, 31) (dual of [262163, 261954, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4209, 262156, F4, 31) (dual of [262156, 261947, 32]-code), using
- net defined by OOA [i] based on linear OOA(4209, 17477, F4, 31, 31) (dual of [(17477, 31), 541578, 32]-NRT-code), using
(181, 181+31, 123383)-Net over F4 — Digital
Digital (181, 212, 123383)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4212, 123383, F4, 2, 31) (dual of [(123383, 2), 246554, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4212, 131084, F4, 2, 31) (dual of [(131084, 2), 261956, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4212, 262168, F4, 31) (dual of [262168, 261956, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4181, 262144, F4, 27) (dual of [262144, 261963, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 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(30) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- OOA 2-folding [i] based on linear OA(4212, 262168, F4, 31) (dual of [262168, 261956, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(4212, 131084, F4, 2, 31) (dual of [(131084, 2), 261956, 32]-NRT-code), using
(181, 181+31, large)-Net in Base 4 — Upper bound on s
There is no (181, 212, large)-net in base 4, because
- 29 times m-reduction [i] would yield (181, 183, large)-net in base 4, but