The United States of America As Represented by the Director of the

The United States of America As Represented by the Director of the National Security Agency

The United States of America As Represented by the National Security

Michael Samuel Postol

A method of lattice-quantizing an eight-long data point to minimize storage requirements by acquiring the data point, multiplying each coordinate of the data point by {square root over (2)} to form an inflated data point, rounding each coordinate of the rounded and inflated data point to the nearest integer, modulo-two reducing each coordinate of the rounded and inflated data point to form an initial codeword, multiplying a parity-check matrix of an eight-bit Extended Hamming Code by the result of the last step to form a syndrome, correcting any single-bit errors and double-bit errors, if any, in the initial codeword and the rounded and inflated data point, transforming the coordinate system of the initial codeword to an extended quadratic residue form, creating a signal packet, and transmitting the signal packet to a receiver. The receiver receives the signal packet, recovers the intended Extended Hamming Code codeword by a reverse transform of lattice coordinates, modifies the codeword according to the signal packet in order to recover the rounded and inflated data point, replaces each 1 in the rounded and inflated data point by 1/{square root over (2)} of the same polarity, and each 2 by {square root over (2)} of the same polarity in order to recover the data point.

Method for storage and reconstruction of the extended hamming code for an 8-dimensional lattice quantizer

A method of lattice quantizing a 24-long data point to minimize storage requirements by acquiring the data point, multiplying each coordinate of the data point by the square root of 8 to form an inflated data point, finding the closest point of the inflated Leech lattice by any method, creating a signal packet, and transmitting the packet to a receiver. The receiver receives the signal packet, recovers the inflated lattice point with the aid of a table of 89 of the 4096 codewords of the extended Golay code, replaces each one in the inflated lattice point by the reciprocal of the square root of 8 of the same polarity, replaces each 2 in the inflated lattice point by 2 divided by the square root of 8 of the same polarity, replaces each 3 in the inflated lattice point by 3 divided by the square root of 8 of the same polarity, replaces each 4 in the inflated lattice point by 4 divided by the square root of 8 of the same polarity, and replaces each 5 in the inflated lattice point by 5 divided by the square root of 8 of the same polarity in order to recover the data point.

Method for efficient storage and transmission of the centroids of a

A method of lattice-quantizing an eight-long data point to minimize storage requirements and computational complexity by acquiring the data point, multiplying each coordinate of the data point by the square root of two to form an inflated data point, rounding each coordinate of the rounded and inflated data point to the nearest integer, reducing modulo-two each coordinate of the rounded and inflated data point to form an initial codeword, multiplying a parity-check matrix of an eight-bit Extending Hamming Code by the result of the last step to form a syndrome, correcting any single-bit errors and double-bit errors, if any, in the initial codeword and the rounded and inflated data point, creating a signal packet if the codeword does not contain any single-bit errors and double-bit errors, and transmitting the signal packet to a receiver. The receiver receives the signal packet, recovers the intended codeword from the table of sixteen codewords in the Extended Hamming Code, modifies the codeword according to the signal packet in order to recover the rounded and inflated data point, replaces each one in the rounded and inflated data point by the reciprocal of the square root of two of the same polarity, and replaces each two in the rounded and inflated data point by the square root of two of the same polarity in order to recover the data point.

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Potomac, MD, United States of America

Michael Samuel Postol