Best Known (123−23, 123, s)-Nets in Base 4
(123−23, 123, 1491)-Net over F4 — Constructive and digital
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
(123−23, 123, 9084)-Net over F4 — Digital
Digital (100, 123, 9084)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4123, 9084, F4, 23) (dual of [9084, 8961, 24]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(4123, 16402, F4, 23) (dual of [16402, 16279, 24]-code), using
(123−23, 123, 7785473)-Net in Base 4 — Upper bound on s
There is no (100, 123, 7785474)-net in base 4, because
- 1 times m-reduction [i] would yield (100, 122, 7785474)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 28 269559 898752 403117 999411 115892 858552 789520 292703 132438 844891 277118 325248 > 4122 [i]