Quantization Codes Comprising Multiple Orthonormal Bases Alexei Ashikhmin

Quantization Codes Comprising Multiple Orthonormal Bases Alexei Ashikhmin Bell Labs § MIMO Broadcast Transmission § Quantizers Q(m) for MIMO Broadcast Systems • transmission to mobiles with orthogonal channel vectors • transmission to mobiles with almost orthogonal channel vectors § Simulation Results § Algebraic Constructions of Q(m)

MIMO Broadcast Transmission Base Station is a quantization code The Base Station (BS): • chooses some mobiles, for example mobiles 1, 2, 3 • forms and using computes a precoding matrix • transmits to mobiles 1, 2, 3 using the precoding matrix

Requirements for a quantization code • should provide good quantization (for given size • should afford a simple decoding • should have many sets of M orthogonal codewords (bases of ) ) is the channel vector of BS is the channel vector of If are pairwise orthogonal then signals sent to not interfere with each other do

Base Station • Mobiles quantize: • Base Station strategy – among find orthogonal codewords, say , and transmit to the corresponding mobiles 1, 3, 5 • The channel vectors of these mobiles will be almost orthogonal

Let us have a quantization code orthogonal codewords If a channel vector and mark is quantized into we say that is occupied by • If the number of mobiles (channel vectors) is large, e. g. , then with a high probability all codewords will be occupied • In this case even if we have only a few sets of orthogonal codewords, we easily find a set of occupied orthogonal codewords

Example • Let and the number of mobiles is small, say • Let • If are many sets of orthogonal code vectors there is a chance to find occupied orthogonal codewords • For example, if are sets of orthogonal codewords. Then

Example: The number of antennas (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) (1, 0, 1, 0), (0, 1, 0, 1), (1, 0, -1, 0), (0, -1, 0, 1) The first code in the family: (1, 0, -i, 0), (0, 1, 0, -i), (1, 0, i, 0), (0, 1, 0, i) (1, 1, 0, 0), (0, 0, 1, 1), (1, -1, 0, 0), (0, 0, -1, 1) (1, -i, 0, 0), (0, 0, 1, -i), (1, i, 0, 0), (0, 0, 1, i) (1, 0, 0, 1), (0, 1, 1, 0), (1, 0, 0, -1), (0, 1, -1, 0) (for practical applications we (1, 0, 0, -i), (0, 1, i, 0), (1, 0, 0, i), (0, 1, -i, 0) add four vectors to the code (1, 1, 1, 1), (1, -1, 1, -1), (1, -1, 1) to make the code size 64) 105 orthogonal bases (1, 1, -i), (1, -i, i), (1, 1, i, i), (1, -1, i, -i) (1, -i, 1, -i), (1, i, 1, i), (1, -i, -1, i), (1, i, -1, -i) (1, -i, -1), (1, i, -i, 1), (1, -i, i, 1), (1, i, i, -1) (1, -i, 1), (1, i, -1), (1, -i, i, -1), (1, i, i, 1) (1, -i, 1, i), (1, i, 1, -i), (1, -i, -1, -i), (1, i, -1, i) (1, 1, 1, -1), (1, -1, 1, 1), (1, 1, -1, 1), (1, -1, -1) (1, 1, -i, i), (1, -i, -i), (1, 1, i, -i), (1, -1, i, i)

The number of mobiles • The bases form the constant weight code (n=60, |C|=105, w=4). • With probability 0. 65 will find four orthogonal occupied codewords • With probability 0. 349 will find three orthogonal occupied codewords

Examples (continued) 1. The number of orthogonal bases is 105. Each codeword belongs to 7 bases. The bases form the constant weight code (n=60, |C|=105, w=4). 2. The number of orthogonal bases is 1076625. Each codeword belongs to 7975 bases. The bases form the constant weight code (n=1080, |C|=1076625, w=8) If K is small that the probability to find M occupied orthogonal codewords is also small What to do? - Use almost orthogonal codewords

Simulation Results All results for M=8, i. e. the number of Base Station antennas is 8 K=1000 Q(3) Yoo and Goldsmith greedy alg. with RVQ with Reg. ZF RVQ with ZF

If K=50 typically we can find 5 or 6 Q(3), occupied codewords Q(3),

Q(3) greedy alg.

Mutually Unbiased Bases (MUB) Def. Orthonormal bases if for any of are mutually unbiased we have Theorem The number of MUBs Def. (i. e. Bases ) is a full size MUB set. form a full size MUB set

• MUB sets form a constant weight code C (n=15, |C|=6, w=5) • If K is small the chance that M occupied codewords are covered by an MUB set is significantly higher than that they are covered by a basis

There are 840 full size MUB sets • Let are orthogonal • Let To transmit efficiently to mobiles with we design a special precoding matrix , each belongs to 56

are orthogonal and are orthogonal Transmission to

Decoding Example M=8 Q(3), |Q(3)|=1080 Random Code C, |C|=1080 • Complex multiplications 0 8*1080 1500 7*1080 • Complex summations

Construction of Q(m) is a code in There are two equivalent methods for construction of Q(m): 1. Group theoretic approach 2. Coding theory approach

Orthogonal Projectors • A subspace projector of can be defined by its orthogonal , i. e. • a • is an orthogonal projector iff

Group Theoretic Construction of Q(m) Pauli matrices: where

It is easy to check that Theorem is an orthogonal projector and

Def. Vectors and are orthogonal (with respect to the symplectic inner product) if • is a set of orthogonal independent vectors • . Lemma 2 The operator and is an orthogonal projector on a subspace ,

It is easy to check that Thus and defines a subspace . therefore So is a line.

Construction of Q(m) • Take all sets of orthogonal independent vectors • Take all choices of • For each set and set defines a line, in other words compute defines a code vector of Q(m).

Coding Theory approach for construction of Q(m) is obtained by merging of 1. Binary Reed-Muller codes RM(r, m); • is the order or RM(r, m), • the code length is 2. Codes B(m) over the alphabet {1, -1, i, -i} • the code length is

Merging RM(r, m) and B(m) into Q(m) r changes from m=2 to 0: 1. r=m=2: take the all minimum weight codewords of RM(2, 2): (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) 2. r=m-1=1: substitute codewords of 3. into the minimum weight codewords of RM(1, 2) Minimum weight codeword of RM(1, 2): (1, i) (1, -i) (1, 1) (1, -1) Codewords of Q(2): (1, i, 0, 0) (0, 1, i, 0) (1, 1, 0, 0) (0, 1, 1, 0) (0, 1, -i, 0) (1, -i, 0, 0) (1, 1, 0, 0) (0, 1, 1, 0) (1, -1, 0, 0) (0, 1, -1, 0) 4. 3. r=m-2=0: take the only minimum weight codeword of RM(r, m)=RM(0, m): (1, 1, 1, 1) and substitute into its nonzero positions codewords of

r=0, minimum weights v codewords of RM(2, 2) (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) (1, 1, 0, 0), (1, i, 0, 0), (1, -1, 0, 0), (1, -i, 0, 0) (1, 0, 1, 0), (1, 0, i, 0), (1, 0, -1, 0), (0, 1, 0, -i) r=1, minimum weights v codewords of RM(1, 2) v +codewords of B(1) (1, 0, 0, 1), (1, 0, 0, i), (1, 0, 0, -1), (1, 0, 0, -i) (0, 1, 1, 0), (0, 1, i, 0), (0, 1, -1, 0), (0, 1, -i, 0) (0, 1, 0, 1), (0, 1, 0, i), (0, 1, 0, -1), (1, 0, -i, 0) (0, 0, 1, 1), (0, 0, 1, i), (0, 0, -1, 1), (0, 0, 1, -i) (1, 1, 1, 1), (1, -1, 1, -1), (1, -1, 1 (1, 1, -i), (1, -i, i), (1, 1, i, i), (1, -1, i, -i), (1, -i, 1, -i), (1, i, 1, i), (1, -i, -1, i), (1, i, -1, -i), r=2, minimum weights v codewords of RM(0, 2) v +codewords of B(2) (1, -i, -1), (1, i, -i, 1), (1, -i, i, 1), (1, i, i, -1), (1, -i, 1), (1, i, -1), (1, -i, i, -1), (1, i, i, 1), (1, -i, 1, i), (1, i, 1, -i), (1, -i, -1, -i), (1, i, -1, i), (1, 1, 1, -1), (1, -1, 1, 1), (1, 1, -1, 1), (1, -1, -1 (1, 1, -i, i), (1, -i, -i), (1, 1, i, -i), (1, -1, i, i)

Theorem Example: Theorem (Inner product distribution of Q(m)). For any we have and the number of such that Example: in Q(2) there are 15 vectors in Q(3) there are 315 vectors is such that

Theorem For any basis such that there exist bases is an MUB set. Theorem The maximum root-mean-square (RMS) inner product is