Best Known (126−23, 126, s)-Nets in Base 4
(126−23, 126, 1491)-Net over F4 — Constructive and digital
Digital (103, 126, 1491)-net over F4, using
- 43 times duplication [i] based on digital (100, 123, 1491)-net over F4, using
- net defined by OOA [i] based on linear OOA(4123, 1491, F4, 23, 23) (dual of [(1491, 23), 34170, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4123, 16402, F4, 23) (dual of [16402, 16279, 24]-code), using
- construction XX applied to Ce(22) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(41, 16, F4, 1) (dual of [16, 15, 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, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(22) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(4123, 16402, F4, 23) (dual of [16402, 16279, 24]-code), using
- net defined by OOA [i] based on linear OOA(4123, 1491, F4, 23, 23) (dual of [(1491, 23), 34170, 24]-NRT-code), using
(126−23, 126, 11077)-Net over F4 — Digital
Digital (103, 126, 11077)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4126, 11077, F4, 23) (dual of [11077, 10951, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4126, 16412, F4, 23) (dual of [16412, 16286, 24]-code), using
- construction XX applied to Ce(22) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-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(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(22) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4126, 16412, F4, 23) (dual of [16412, 16286, 24]-code), using
(126−23, 126, large)-Net in Base 4 — Upper bound on s
There is no (103, 126, large)-net in base 4, because
- 21 times m-reduction [i] would yield (103, 105, large)-net in base 4, but