Method for efficient storage and transmission of the centroids of a leech lattice quantizer
First Claim
1. A method of lattice-quantizing and communicating 24 dimensional data points in a manner that minimizes storage requirements, comprising the steps of:
- a) creating an indexed list of 89 codewords, of 24 bits each, from the extended Golay code, from which all codewords of the extended Golay code can be produced through the circular rotation of the first 23 bits of each of said 89 codewords and the logical complementing of each of said possible circular rotations of each of said 89 codewords, and indexing them by codeword position numbers from 1 to 89;
b) forming an inflated Leech lattice by multiplying each lattice point of a Leech lattice by the square-root of 8;
c) acquiring 24-dimensional data points in sequence; and
d) for each of said 24-dimensional data points acquired in sequence, lattice-quantizing each of said 24-dimensional data points to create a signal packet and transmitting said signal packet to a receiver, further comprising the steps of;
i) multiplying each coordinate of said 24-dimensional data point acquired in sequence by the square root of 8 to form an inflated data point;
ii) finding the closest lattice point to said inflated data point on said inflated Leech lattice;
iii) determining the shell number and type number of said closest lattice point on said inflated Leech lattice;
iv) for said closest lattice points in shell 2 and of type 3, assembling a signal packet for a lattice point of type 3 in shell 2, having 1 bit to indicate the point is in shell 2, 2 bits to indicate the point is of type 3, 5 bits to indicate the position of the first positive or negative 4, 5 bits to indicate the position of the second positive or negative 4, 1 bit to indicate the polarity of the first 4, and having 1 bit to indicate the polarity of the second 4;
v) for said closest lattice points not in shell 2 and of type 3, determining the parameters for a signal packet, further comprising the steps of;
determining the C-set of said closest lattice point;
forming a closest lattice point extended Golay codeword from said C-set by substituting 1s in the positions of said C-set and 0s elsewhere;
logically complementing said closest lattice point extended Golay codeword if its weight is greater than 12 or if its weight is equal to 12 and it has a final bit of 1;
circularly shifting the first 23 bits of said closest lattice point extended Golay codeword until said closest lattice point extended Golay codeword is the smallest binary number closest lattice point extended Golay codeword achievable through such circular shifting;
for said smallest binary number closest lattice point extended Golay codeword that is not all-zeros, finding said closest lattice point extended Golay codeword in said indexed list of 89 codewords and recording the indexed codeword position number; and
for said smallest binary number closest lattice point extended Golay codeword that is all-zeros, setting the codeword position number to 0;
vi) for said closest lattice points not in shell 2 and of type 3, constructing a signal packet identifying the shell and type of said closest lattice point, smallest binary number closest lattice point extended Golay codeword, whether said smallest binary number closest lattice point extended Golay codeword was logically complemented, and the position and polarity of the positions of said C-set; and
vii) transmitting said signal packet to a receiver.
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Abstract
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.
9 Citations
9 Claims
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1. A method of lattice-quantizing and communicating 24 dimensional data points in a manner that minimizes storage requirements, comprising the steps of:
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a) creating an indexed list of 89 codewords, of 24 bits each, from the extended Golay code, from which all codewords of the extended Golay code can be produced through the circular rotation of the first 23 bits of each of said 89 codewords and the logical complementing of each of said possible circular rotations of each of said 89 codewords, and indexing them by codeword position numbers from 1 to 89; b) forming an inflated Leech lattice by multiplying each lattice point of a Leech lattice by the square-root of 8; c) acquiring 24-dimensional data points in sequence; and d) for each of said 24-dimensional data points acquired in sequence, lattice-quantizing each of said 24-dimensional data points to create a signal packet and transmitting said signal packet to a receiver, further comprising the steps of; i) multiplying each coordinate of said 24-dimensional data point acquired in sequence by the square root of 8 to form an inflated data point; ii) finding the closest lattice point to said inflated data point on said inflated Leech lattice; iii) determining the shell number and type number of said closest lattice point on said inflated Leech lattice; iv) for said closest lattice points in shell 2 and of type 3, assembling a signal packet for a lattice point of type 3 in shell 2, having 1 bit to indicate the point is in shell 2, 2 bits to indicate the point is of type 3, 5 bits to indicate the position of the first positive or negative 4, 5 bits to indicate the position of the second positive or negative 4, 1 bit to indicate the polarity of the first 4, and having 1 bit to indicate the polarity of the second 4; v) for said closest lattice points not in shell 2 and of type 3, determining the parameters for a signal packet, further comprising the steps of; determining the C-set of said closest lattice point; forming a closest lattice point extended Golay codeword from said C-set by substituting 1s in the positions of said C-set and 0s elsewhere; logically complementing said closest lattice point extended Golay codeword if its weight is greater than 12 or if its weight is equal to 12 and it has a final bit of 1; circularly shifting the first 23 bits of said closest lattice point extended Golay codeword until said closest lattice point extended Golay codeword is the smallest binary number closest lattice point extended Golay codeword achievable through such circular shifting; for said smallest binary number closest lattice point extended Golay codeword that is not all-zeros, finding said closest lattice point extended Golay codeword in said indexed list of 89 codewords and recording the indexed codeword position number; and for said smallest binary number closest lattice point extended Golay codeword that is all-zeros, setting the codeword position number to 0; vi) for said closest lattice points not in shell 2 and of type 3, constructing a signal packet identifying the shell and type of said closest lattice point, smallest binary number closest lattice point extended Golay codeword, whether said smallest binary number closest lattice point extended Golay codeword was logically complemented, and the position and polarity of the positions of said C-set; and vii) transmitting said signal packet to a receiver. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9)
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Specification