Electrical computers: arithmetic processing and calculating – Electrical digital calculating computer – Particular function performed
Reexamination Certificate
2004-06-10
2009-10-20
Malzahn, David H (Department: 2193)
Electrical computers: arithmetic processing and calculating
Electrical digital calculating computer
Particular function performed
Reexamination Certificate
active
07606847
ABSTRACT:
We describe a method for dense encoding of information. Bennet and Wiesner (Phys. Rev. Lett. 69:2881-2884, 1992), using EPR-pairs, showed that n bits can be encoded by n/2 quantum-bits, from which the original bits can be retrieved. Here, in a completely different (non-quantum) setting, we give a method for more dense encoding: In our method n bits x1, x2, . . . , xnare mapped by a linear transform B over the 6-element ring Z6to numbers z1, z2, . . . , ztfrom ring Z6with t=no(1)(i.e., much fewer numbers) (Quantity o(1) here denotes a positive number which goes to 0 as n goes to infinity), then, by applying another linear transform C to these zi's, we will get back n elements of ring Z6, x′1, x′2, . . . , x′n, where, e.g., x′1may have the form x′1=x1+3x2+4x3. One can get back x1simply by running through the values of xion the set 0, 1, 2, 3, 4, 5, and noticing that only x1has period 6, (3x2has period 2, 4x3has period 3). Our results generalize for any non-prime-power composite number m instead of 6. We also apply this method for fast computation of matrix multiplication and for compacting and extending matrices with linear transforms.
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