Best Known (88, 108, s)-Nets in Base 4
(88, 108, 1640)-Net over F4 — Constructive and digital
Digital (88, 108, 1640)-net over F4, using
- net defined by OOA [i] based on linear OOA(4108, 1640, F4, 20, 20) (dual of [(1640, 20), 32692, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4108, 16400, F4, 20) (dual of [16400, 16292, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4108, 16401, F4, 20) (dual of [16401, 16293, 21]-code), using
- construction XX applied to Ce(20) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- 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(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(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(20) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4108, 16401, F4, 20) (dual of [16401, 16293, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4108, 16400, F4, 20) (dual of [16400, 16292, 21]-code), using
(88, 108, 9534)-Net over F4 — Digital
Digital (88, 108, 9534)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4108, 9534, F4, 20) (dual of [9534, 9426, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4108, 16401, F4, 20) (dual of [16401, 16293, 21]-code), using
- construction XX applied to Ce(20) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- 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(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(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(20) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4108, 16401, F4, 20) (dual of [16401, 16293, 21]-code), using
(88, 108, 4798464)-Net in Base 4 — Upper bound on s
There is no (88, 108, 4798465)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 105312 498920 240786 702341 324858 351761 444553 485548 047388 114493 948928 > 4108 [i]