MinT - Best Known (123−21, 123, s)-Nets in Base 8

Best Known (123−21, 123, s)-Nets in Base 8

(123−21, 123, 52429)-Net over F₈ — Constructive and digital

Digital (102, 123, 52429)-net over F₈, using

8¹ times duplication [i] based on digital (101, 122, 52429)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹²², 52429, F₈, 21, 21) (dual of [(52429, 21), 1100887, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(8¹²², 524291, F₈, 21) (dual of [524291, 524169, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(8¹²², 524294, F₈, 21) (dual of [524294, 524172, 22]-code), using
      - trace code [i] based on linear OA(64⁶¹, 262147, F₆₄, 21) (dual of [262147, 262086, 22]-code), using
        constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
        linear OA(64⁶¹, 262144, F₆₄, 21) (dual of [262144, 262083, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(64⁵⁸, 262144, F₆₄, 20) (dual of [262144, 262086, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(123−21, 123, 524296)-Net over F₈ — Digital

Digital (102, 123, 524296)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²³, 524296, F₈, 21) (dual of [524296, 524173, 22]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(8¹²², 524294, F₈, 21) (dual of [524294, 524172, 22]-code), using
    - trace code [i] based on linear OA(64⁶¹, 262147, F₆₄, 21) (dual of [262147, 262086, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
        linear OA(64⁶¹, 262144, F₆₄, 21) (dual of [262144, 262083, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(64⁵⁸, 262144, F₆₄, 20) (dual of [262144, 262086, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
  2. linear OA(8¹²², 524295, F₈, 20) (dual of [524295, 524173, 21]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 8¹²²Â > V_b^sâˆ’1(kâˆ’1)Â = 440100 240206 448607 498792 836001 080175 872434 311946 680993 171096 544036 275548 479054 611874 954666 715087 871828 099072 [i]
  3. linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(123−21, 123, large)-Net in Base 8 — Upper bound on s

There is no (102, 123, large)-net in base 8, because

19 times m-reduction [i] would yield (102, 104, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (123−21, 123, s)-Nets in Base 8

(123−21, 123, 52429)-Net over F8 — Constructive and digital

(123−21, 123, 524296)-Net over F8 — Digital

(123−21, 123, large)-Net in Base 8 — Upper bound on s

(123−21, 123, 52429)-Net over F₈ — Constructive and digital

(123−21, 123, 524296)-Net over F₈ — Digital