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| In [[information theory]], the '''bar product''' of two [[linear code]]s ''C''<sub>2</sub> ⊆ ''C''<sub>1</sub> is defined as
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| :<math>C_1 \mid C_2 = \{ (c_1\mid c_1+c_2) : c_1 \in C_1, c_2 \in C_2 \}, </math>
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| where (''a'' | ''b'') denotes the concatenation of ''a'' and ''b''. If the [[code word]]s in ''C''<sub>1</sub> are of length ''n'', then the code words in ''C''<sub>1</sub> | ''C''<sub>2</sub> are of length 2''n''.
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| The bar product is an especially convenient way of expressing the [[Reed–Muller code|Reed–Muller]] RM (''d'', ''r'') code in terms of the Reed–Muller codes RM (''d'' − 1, ''r'') and RM (''d'' − 1, ''r'' − 1).
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| The bar product is also referred to as the | '''''u''''' | '''''u'''''+'''''v''''' | construction<ref>{{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthors=N.J.A. Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | year=1977 | isbn=0-444-85193-3 | page=76 }}</ref>
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| or ('''''u''''' | '''''u''''' + '''''v''''') construction.<ref>{{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2nd ed | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=86 | year=1992 | isbn=3-540-54894-7 | page=47 }}</ref>
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| ==Properties==
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| ===Rank===
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| The [[dimension (vector space)|rank]] of the bar product is the sum of the two ranks:
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| :<math>\operatorname{rank}(C_1\mid C_2) = \operatorname{rank}(C_1) + \operatorname{rank}(C_2)\,</math>
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| ====Proof====
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| Let <math> \{ x_1, \ldots , x_k \} </math> be a basis for <math>C_1</math> and let <math>\{ y_1, \ldots , y_l \} </math> be a basis for <math>C_2</math>. Then the set
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| <math>\{ (x_i\mid x_i) \mid 1\leq i \leq k \} \cup \{ (0\mid y_j) \mid 1\leq j \leq l \} </math>
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| is a basis for the bar product <math>C_1\mid C_2</math>. | |
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| ===Hamming weight===
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| The [[Hamming weight]] ''w'' of the bar product is the lesser of (a) twice the weight of ''C''<sub>1</sub>, and (b) the weight of ''C''<sub>2</sub>:
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| :<math>w(C_1\mid C_2) = \min \{ 2w(C_1) , w(C_2) \}. \,</math>
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| ====Proof==== | |
| For all <math>c_1 \in C_1</math>,
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| :<math>(c_1\mid c_1 + 0 ) \in C_1\mid C_2</math>
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| which has weight <math>2w(c_1)</math>. Equally
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| :<math> (0\mid c_2) \in C_1\mid C_2</math>
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| for all <math>c_2 \in C_2 </math> and has weight <math>w(c_2)</math>. So minimising over <math>c_1 \in C_1, c_2 \in C_2</math> we have
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| :<math>w(C_1\mid C_2) \leq \min \{ 2w(C_1) , w(C_2) \} </math>
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| Now let <math>c_1 \in C_1</math> and <math>c_2 \in C_2</math>, not both zero. If <math>c_2\not=0</math> then:
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| : <math> | |
| \begin{align}
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| w(c_1\mid c_1+c_2) &= w(c_1) + w(c_1 + c_2) \\
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| & \geq w(c_1 + c_1 + c_2) \\
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| & = w(c_2) \\
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| & \geq w(C_2)
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| \end{align}
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| </math>
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| If <math>c_2=0</math> then
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| : <math>\begin{align} | |
| w(c_1\mid c_1+c_2) & = 2w(c_1) \\
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| & \geq 2w(C_1)
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| \end{align}
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| </math>
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| so
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| :<math>w(C_1\mid C_2) \geq \min \{ 2w(C_1) , w(C_2) \} </math>
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| ==See also==
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| * [[Reed–Muller code]]
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| ==References==
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| <references />
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| {{DEFAULTSORT:Bar Product (Coding Theory)}}
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| [[Category:Information theory]]
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| [[Category:Coding theory]]
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