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Application of Special Matrices in Multi-User Information Theory: Broadcast and Switching Channels

This text discusses the application of special rank ( k ) matrices in multi-user information theory, specifically focusing on their role in broadcast and switching channels. It explores coding strategies for memories with defects, providing the existence proof for uniform rank ( k ) matrices and their implications for efficiently transmitting information. Detailed examples illustrate how these matrices can retrieve lost sequences and efficiently handle channel defects, emphasizing their utility in modern coding theory and practical communication systems.

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Application of Special Matrices in Multi-User Information Theory: Broadcast and Switching Channels

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  1. Information theory Multi-user information theory Part 7: A special matrix application A.J. Han Vinck Essen, 2002

  2. content • a special rank k x n natrix • the application in • Broadcast channel • Switching channel • Coding for memories with defects • existance proof

  3. Definition: of a uniform rank k matrix • a binary uniform rank k, k x n matrix U - has rank k - after deleting  (n-k) columns the rank of the remaining matrix = k n k deleted

  4. Application (1): the switching channel X1={0,1} Y ={,0,1} X2={0,1} 1 1 1 0 1 0 1 1 0 1 1 = X1 X2 = (01...1) U Result:Y = X2 x U with positions erased () by X1 Sum Rate: k/n + h(k/n)

  5. Continuation: Why? X2 can be retrieved from the remaining part (rank = k i.e. An inverse exists) transmitted k bits X1 specifies ~ 2nh([ n-k)/n]) = 2nh(1-k)n) sequences transmitted nh(1-k/n) = nh(k/n) bits

  6. Application (2): the broadcast channel Z X Y 0 0 0 1 0 1 2 1 1 Step 1: encode information for y Y has a maximum of k zeros Y = ( 1 0 1 0 1 1) C(y)= (1/2, 0, 1/2, 0, 1/2, 1/2)

  7. Application (2): the broadcast channel k-zeros Y = ( 1 0 1 1 0 1 1 ) X=(X1, X2 , Xn-k) C(X) = ( 0 00 1 10 1 ) C(X,Y) = ( 1 0 0 0 1 0 0 ) C(X)  C(X,Y) = ( 1 0 0 1 0 0 1 ) Z = ( 2 0 1 2 0 1 2 )   Property: Z has the same zeros as C(y)

  8. Application (2): the broadcast channel Z = ( 2 0 1 2 0 1 2 )  y = ( 1 0 1 1 0 1 1 )  C(X)  C(X,Y) = ( 1 0 0 1 0 0 1 )  C(X,Y) = ( 1 0 0 0 1 0 0 )  C(X) = ( 0 0 0 1 1 0 1 )

  9. Continuation: Why does it work? First k bits of C(X,Y) uniquely determine C(X,Y) U = C(X,Y)  C(X,Y)  C(X) ( ? ? ? ) C(X) = 00000 X1X2 Xn-k  no influence first k bits Any pattern of k bits can be constructed s.t. C(X)  C(X,Y) has zeros where Y has

  10. Transmitted information: n-k bits with C(X) n h( k/n)= nh((n-k)/n) bits with Y Hence: efficiency per transmission (n-k)/n + h((n-k)/n)

  11. Memory with defects: Y specifies a vector with  k defects Y = ( **0**0*1**1****1*) C(X) = ( 000000 X1X2 Xn-k ) Store: C(X)  C(X,Y) matches the defects in Y Read: C(X)  C(X,Y) errorfree and add C(X,Y) to get C(X) Efficiency:= 1 - k/n !

  12. Excercise: Give the matrix U and efficiency for k = 1 k = 2 k = n-1

  13. Existance (1) Ingredients: specify (n-k) erased columns Property: remaining part of G has rank k

  14. Existance (2) Y = # different patterns of (n-k) erased columns X = # of possible rank k matrices for a specific pattern k n-k k y X   total number of matrices = 2kn One matrix must have more than entries

  15. Existance (3) • # different patterns of column erasure Y ~ • # of invertible k x k matrices F= (2k–1)(2k–2)•••(2k–2k-1) • A specified pattern allows X = 2(n-k)k F matrices G • 2(n-k)k F  cF 2nk where cF = 0.28

  16. Existance (4) Average # of allowed patterns per matrix Conclusion: there exists at least one (k x n) matrix for which  different patterns of up to (n-k) column erasures leave a matrix of rank k =

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