# US 12,113,610 B1 · User-distinguished finite-field resource construction method and finite-field multiple access system

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- Number: US 12,113,610 B1
- Document ID: US12113610B1
- Status: Granted
- Kind: B1
- Application number: 18/612,803
- Filed: 2024-03-21
- Published: 2024-09-26
- Granted: 2024-10-08
- Earliest claimed: 2023-03-21
- Assignees: Harbin Institute of Technology
- Applicants: Harbin Institute of Technology
- Inventors: Qiyue Yu; Jiangxuan Li; Shu Lin
- Web page: https://patentgrep.com/patent/US12113610B1

## CPC

- H04J 13/12: Generation of orthogonal codes
- H04J 13/16: Code allocation
- Y02D 30/70: in wireless communication networks
- H04J: MULTIPLEX COMMUNICATION (transmission in general H04B; peculiar to transmission of digital information H04L5/00; systems for the simultaneous or sequential transmission of more than one television signal H04N7/08; in exchanges H04Q11/00; stereophonic systems H04S)
- Y02D: CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE

## Abstract

The present disclosure relates to the field of communication technologies and in particular to a user-distinguished finite-field resource construction method and a finite-field multiple access system. In order to solve the problem of the limitation of the multiple access resource in the current communication field, the present disclosure employs a user-distinguished finite-field resource construction method to construct a basic-field resource and/or extension-field resource, i.e. finite-field resource. During the use of the finite-field resource, each user sending a binary sequence is assigned one codebook marking symbols that 0 and 1 are respectively mapped into a finite field. The transmitter sends a corresponding finite-field symbol sequence. At the receiver, based on the received finite-field symbols, a finite-field symbol sent by each user can be determined uniquely and thus, a binary symbol sent by each user can be decoded. The present disclosure is applied to the finite-field multiple access system.

## Claims

### Claim 1

A user-distinguished finite-field resource construction method, executed by a processor, wherein the construction method is a basic-field resource construction method and comprises the following steps:
let GF(p) be a prime field, and p−1 nonzero elements in the field are divided into (p−1)/2 mutually-disjoint element pairs EP, wherein each EP comprises two elements, k and additive inverse p−k in the field, and the element pair (k, p−k) is referred to as additive inverse element pair AIEP; such division is denoted as P;
let J be a positive integer not greater than (p−1)/2, namely, 1≤J≤(p−1)/2; let C1,C2, . . . , CJ be J AIEPs in the P, and thus Cj=(tj,p−tj); wherein 1≤j≤J, 1≤tj≤p; tj is a nonzero element in GF(p), and C refers to a set {C1,C2, . . . , CJ};
let (u1,u2, . . . , uJ) be one J tuple on GF(p), wherein uj is one element in Cj; and thus the J tuple (u1,u2, . . . , uJ) is one element in Cartesian product C1×C2× . . . ×CJ; the mod p of each element and τ=⊕j=1Juj in the tuple are referred to as finite-field sum-pattern (FFSP) of the J tuple; wherein ⊕ represents modular addition operation;
the J tuple (p−u1, p−u2, . . . , p−uJ) is also one element in Cartesian product and its sum pattern is p−τ=p−⊕j=1Juj; if the sum pattern of the (u1,u2, . . . , uJ) is 0, the sum pattern of the (p−u1, p−u2, . . . , p−uJ) is also 0;
let (u1,u2, . . . , uJ) and (u′1,u′2, . . . , u′J) be two J tuples in C1×C2× . . . ×CJ; if ⊕j=1Juj≠⊕j=1Ju′j, one sum pattern uniquely specifies one J tuple in C1×C2× . . . ×Cj, namely,
(
u
1
,
u
2
,
…
,
u
J
)
↔
⊕
j
=
1
J
u
j
is one one-to-one mapping;
given one sum pattern τ=⊕j=1Juj, a corresponding J tuple (u1,u2, . . . , uJ) is uniquely restored; the C1×C2× . . . ×CJ is said to have a unique sum pattern mapping structure USPM, and J≤log2(p−1);
for the set C={C1,C2, . . . , CJ} formed by J AIEPs, if the set has a USPM structure, the set serves as a finite-field resource in multiple access communication;
in one multiple access communication system, J AIEPs are assigned to J users; if each user respectively transmits a corresponding AIEP symbol, a transmitter collects J transmission symbols, obtains finite-field sum-pattern (FFSP) τ=⊕j=1Juj and sends the sum pattern over a multiple access channel; the J transmission symbols are restored uniquely and unambiguously by the sum pattern τ;
if each J tuple is seen as codewords of J users, the set C=C1×C2× . . . ×CJ forms one J user code totaling 2J codewords in GF(p); C=C1×C2× . . . ×CJ is said to be one AIEP code uniquely decodable by the J users in GF(p), which is denoted as J user UDAIEP code; when this code is used in a J user multiple access communication system, the symbol sent by the j-th user corresponds to the j-th component ui of the codewords.

### Claim 2 (depends on 1)

A user-distinguished finite-field resource construction method, wherein the construction method is an extension-field resource construction method, and comprises the following steps:
based on the basic-field resource constructed in claim 1, GF(pm) is an extension field of the GF(p), m is an integer, and an m-order primitive polynomial of the GF(pm) is expressed as:
g(X)=g0+g1X+gX2+ . . . +gmXm
in this field, there are a total of pm elements, and g0, g1, . . . , gm are elements in the GF(p);
let α be a primitive element of the GF(pm); thus, the elements in the field are represented as 0,1,α,α2, . . . , αpm−2; each element is expressed by a linear combination of α0=1,α,α2, . . . αm−1, and coefficients of the linear combinations all belong to the GF(p), namely,
αj=αj,0+αj,1α+αj,2α2+ . . . +αj,m−1αm−1
wherein the coefficient of the linear combination αj=αj,0, αj,1, αj,2, . . . , αj,m−1 is an element in the GF(p);
from the above formula, it is seen that αj is expressed uniquely by the m tuple (αj,0, αj,1, . . . , αj,m−1); thus, a sum of two elements in the extension field is expressed as:
αj+αk=(αj,0+αk,0)+(αj,1+αk,1)α+ . . . +(αj,m−1+αk,m−1)αm−1
the corresponding m tuple is expressed as:
((αj,0+αk,0),(αj,1+αk,1), . . . (αj,m−1+αk,m−1))
for 0≤i≤m, if (αj,i,αk,i) is an additive inverse element pair on the GF(p), namely, αj,i+αk,i=0, then, αj+αk=0, namely, (αj,αk) is one AIEP on the GF(pm); if (αj,i,αk,i) is a nonzero AIEP on the GF(p), then αi(αj,i,αk,i)□(αj,iαi,αk,iαi) is one AIEP on the GF(pm);
let C=(C1,C2, . . . , CJ) be an AIEP set on the GF(p), and have a USPM structure; let Cj=(tj, p−tj), and use ψi(tj, p−tj) to represent an AIEP on the GF(pm) :
αiCj=(tjαi, (p−tj)αi), thus, Ψi={ψi(t1, p−t1), ψi(t2, p−t2), . . . , ψi(tJ, p−tJ} is an AIEP set with the USPM structure on the GF(pm); the Cartesian product of the elements in the Ψi forms a J user UDAIEP code containing 2J codewords on the GF(pm); for i=0,1, . . . , m−1, m J user UDAIEP codes are constructed, and Ψ0,Ψ1, . . . , Ψm−1 are mutually disjoint;
the elements in the AIEP pair ψi(tj, p−tj)=(tjαi,(p−tj)αi are expressed by the m tuple in the following form; in the m tuple, tj and p−tj appear at the i-th position with other positions being zero, namely:
ψi(t1, p−t1)=((0,0, . . . , t1, 0, . . . , 0), (0, 0, . . . , p−t1, 0, . . . , 0))
ψi(t2, p−t2)=((0,0, . . . , t2, 0, . . . , 0), (0, 0, . . . , p−t2, 0, . . . , 0))
.
.
.
ψi(tm, p−tm)=((0,0, . . . , tm, 0, . . . , 0), (0, 0, . . . , p−tm, 0, . . . , 0))
wherein Ψ0,Ψ1, . . . , Ψm−1 defines m orthogonal UDAIEPs corresponding to J users;
Ψ□Ψ0∪Ψ1∪ . . . ∪Ψm−1 forms one Jm user orthogonal UDAIEP code containing 2Jm codewords on the GF(pm) .

### Claim 3 (depends on 2)

A finite-field multiple access system, wherein a finite-field resource in the system is an extension-field resource constructed using the user-distinguished finite-field resource construction method of claim 2;
a downlink in the system uses a downlink transmission method using a finite-field resource to perform downlink transmission; an uplink uses an uplink transmission method using a finite-field resource to perform uplink transmission;
the downlink transmission method using the finite-field resource uses as a finite field the basic-field resource constructed using the user-distinguished finite-field resource construction method or the extension-field resource constructed using the user-distinguished finite-field resource construction method to achieve downlink transmission, which comprises the following specific steps:
the basic-field resource or extension-field resource of the finite field is used as a multiple access resource, the finite field is GF(p), and p>2; thus, the number of users supported by a finite-field multiple access system is J≤└log 2(p−1)┘, and based on the process of constructing the basic-field resource or extension-field resource, the J user UDAIEP code, i.e. C1,C2, . . . , CJ is constructed as a multiple access resource and the user j is assigned a codebook Cj;
let k be a positive number, and bj=(bj,0,bj,1, . . . , bj,k, . . . bj,K−1) is a raw information sequence sent to the j-th user, where bj,k∈GF(2); during sending, one UDAIEP, i.e. Cj, is assigned to the user j, the bj,k is mapped into uj,k, wherein uj,k∈Cj; the mapping is a mapping from a binary field to a non-binary field; this process is expressed using the function FB2NB, i.e. uj,k=FB2NB(bj,k); the sequence bj sent to the user j is mapped to generate a sequence denoted as uj=(uj,0,uj,1, . . . , uj,k, . . . , uj,K), and then the J tuple (u0,k, u1,k, . . . , uJ,k) is one codeword of the J user UDAIEP code;
J sequences sent by performing multiplexing: symbols on the corresponding positions of multiple user information sequences are added up on the GF(p); an obtained result is denoted as u=(u0,u1, . . . , uk, . . . uK−1), wherein uk=⊕j=1Juj,k; the multiplexed sequence u is finite-field sum-pattern (FFSP) of the J user UDAIEP code C;C is uniquely decodable, and the J sequences u1,u2, . . . , uJ are decoded symbol-wise from u without ambiguity, and the process of decoding the J sequences symbol-wise is denoted as FNB2B;
in a downlink system, a sequence w prior to channel encoding is obtained based on the sequence u, and a (N, K) linear block code on GF(Q) is used as a channel encoding v; then the symbols are mapped to a complex field, and then OFDM transmission process is performed;
finally, the sequence of each user is decoded symbol-wise by using an OFDM receiver;
the uplink transmission method using the finite-field resource uses as a finite field the extension-field resource constructed using the user-distinguished finite-field resource construction method to achieve uplink transmission, which comprises the following specific steps:
the extension-field resource is used as a multiple access resource, and based on GF(2m), the basic field is GF(2), which only has one element pair (0,1); using the element pair, m orthogonal UDEPs are constructed on the GF(2m) :
Ψ={ψ0(0,1),ψ1(0,1), . . . , ψi(0,1), . . . , ψm−1(0,1)}
if the system supports at most J users, the user J is assigned EP ψj−1(0,1);
let K be an integer, and bj=(bj,0,bj,1, . . . , bj,k, . . . , bj,K−1) is a binary information sequence sent by the user j; the transmitter firstly maps the sequence symbol-wise from the GF(2) to the non-binary field GF(2m) to generate a sequence uj, i.e. uj,k=FB2NB(bj,k), wherein uj,k∈GF(2m); this mapping relationship is determined by EP ψj−1(0,1) assigned to the user j, and uj,k is expressed by a binary m tuple, uj,k=(uj,k,0,uj,k,1, . . . , uj,k,i, . . . , uj,k,m−1), wherein,
u
j
,
k
,
i
=
{
b
j
,
k
,
i
=
j
-
1
0
,
i
≠
j
-
1
by adding the (N, K) linear block code W, channel encoding is performed on the sequence uj to generate a sequence vj, wherein W is a code on the GF(2m), and a generated matrix G is K×N dimensions; after encoding, there is:
vj=ujG=(νj,0,νj,1, . . . , νj,n, . . . , νj,N−1)
and νj,n∈GF(2m); and νj,n is expressed by using a binary m tuple νj,n=(νj,n,0,νj,n,1, . . . , νj,n,i, . . . , νj,n,m−1);
u=⊕j=1Juj is defined as finite-field sum-pattern (FFSP) of the raw information sequence, and u=(u0,u1, . . . , uk, . . . , uK−1), uk=⊕j=1Juj,k; v=⊕j=1Jvj is defined as finite-field sum-pattern (FFSP) of the encoded information sequence, and v=(ν0,ν1, . . . . , νn, . . . , νN−1), wherein νn=⊕j=1Jνj,n, v=uG;
then, modulation is performed using BPSK to obtain a complex-field vector xj∈□1×Nm, xj=(xj,0,xj,1, . . . , xj,n, . . . , xj,N−1), xj,n=(xj,n,0,xj,n,1, . . . , xj,n,i, . . . , xj,n,m−1), and xj,n,1=2νj,n,i−1; this mapping relationship is denoted as FF2C, that is, a mapping from the finite field to a complex-field; then xj is sent to AWGN multiple access channel;
at the receiver, the received complex-field signal vector y∈□1×Nm is a result of superimposition of the signals sent by J users through the channel, namely,
y
=
∑
j
=
1
J
x
j
+
z
=
r
+
z
wherein z is noise, r is finite-field sum-pattern (FFSP) of the complex-field modulation signal sequence x1,x2, . . . , XJ, which is composed of N m tuples, and written as r=(r0,r1, . . . , rn, . . . , rN−1), wherein rn=(rn,0,rn,1, . . . , rn,i, . . . , rn,m−1);
r
n
,
i
=
∑
j
=
1
J
x
j
,
n
,
i
=
2
⁢
∑
j
=
1
J
x
j
,
n
,
i
-
J
in the first step of decoding reception, by using a conversion function FC2F from the complex-field to the finite field, the complex-field signal is converted into a vector on the GF(2m), FC2F(r)=v=(ν0,ν1, . . . , νn, . . . , νN−1) is composed of N m tuples, and the n-th element is expressed as νn=(νn,0,νn,1, . . . , νn,i, . . . , νn,m−1), wherein,
v
n
,
i
=
F
C
⁢
2
⁢
F
(
r
n
,
i
)
=
⊕
J
j
=
1
v
j
,
n
,
i
next, ŷ=FC2F(r)+z is constructed and then input into a decoder decoding restoration sequence {circumflex over (v)} of the channel encoding W;
an on-field inverse mapping FNB2B is made by the part corresponding to the transmitter to restore the raw sequence {circumflex over (b)}1,{circumflex over (b)}2, . . . , {circumflex over (b)}J sent by each user.

### Claim 4 (depends on 3)

The finite-field multiple access system of claim 3, wherein in the downlink transmission method using the finite-field resource, a process of obtaining the sequence w prior to channel encoding based on the sequence u comprises:
if p and Q are unequal, mapping the sequence u symbol-wise to the sequence w and then performing encoding;
if p and Q are equal, directly using the sequence u as the sequence w and then performing encoding is.

### Claim 5 (depends on 4)

The finite-field multiple access system of claim 4, wherein in the downlink transmission method using the finite-field resource, when p and Q are unequal, t=┌log2(p−1)┐ and Q=2t are set.

### Claim 6 (depends on 3)

The finite-field multiple access system of claim 3, wherein in the downlink transmission method using the finite-field resource, symbol-wise decoding the sequence of each user by using the OFDM receiver comprises the following steps:
the receiver is a reverse process: firstly, a guard interval is removed, and then FFT transform, FDE and parallel-serial conversion are performed to obtain a complex field signal y, and then a restored sequence {circumflex over (v)} is obtained through transform from the complex field to the finite field, and then a sequence ŵ is obtained through channel decoding, and then the sequence is symbol-wise mapped from the GF(Q) to the GF(q), which is completely opposite to the mapping of the transmitter; further, a sequence û is obtained; if û=u, the sequence of each user is decoded symbol-wise.

### Claim 7 (depends on 3)

The finite-field multiple access system of claim 3, wherein in the uplink transmission method using the finite-field resource, the FC2F has the following characteristics:
the value of rn,i obtained by BPSK modulation is: Ω={−J,−J+2, . . . , J−2,J};
the value of νn,i∈GF(2) is only (0)2 or (1)2 corresponding to the value Ω of the rn,i, the value of the νn,i is Ωv={0,1,0,1. . . }, that is: if rn,i=−J, then νn,i=0; if rn,i=−J+2, then νn,i=1.

### Claim 8 (depends on 3)

The finite-field multiple access system of claim 3, wherein in the uplink transmission method using the finite-field resource, in a process that ŷ is constructed and then input into the decoder decoding restoration sequence {circumflex over (v)} of the channel encoding W, if soft decoding is used, a prior probability of the rn,i prior probability distribution is
Pr={CJ0/2J,CJ1/2J, . . . , CJJ−1/2J,CJJ/2J}.

### Claim 9 (depends on 4)

The finite-field multiple access system of claim 4, wherein in the downlink transmission method using the finite-field resource, symbol-wise decoding the sequence of each user by using the OFDM receiver comprises the following steps:
the receiver is a reverse process: firstly, a guard interval is removed, and then FFT transform, FDE and parallel-serial conversion are performed to obtain a complex field signal y, and then a restored sequence {circumflex over (v)} is obtained through transform from the complex field to the finite field, and then a sequence ŵ is obtained through channel decoding, and then the sequence is symbol-wise mapped from the GF(Q) to the GF(q), which is completely opposite to the mapping of the transmitter; further, a sequence û is obtained; if û=u, the sequence of each user is decoded symbol-wise.

### Claim 10 (depends on 5)

The finite-field multiple access system of claim 5, wherein in the downlink transmission method using the finite-field resource, symbol-wise decoding the sequence of each user by using the OFDM receiver comprises the following steps:
the receiver is a reverse process: firstly, a guard interval is removed, and then FFT transform, FDE and parallel-serial conversion are performed to obtain a complex field signal y, and then a restored sequence {circumflex over (v)} is obtained through transform from the complex field to the finite field, and then a sequence ŵ is obtained through channel decoding, and then the sequence is symbol-wise mapped from the GF(Q) to the GF(q), which is completely opposite to the mapping of the transmitter; further, a sequence û is obtained; if û=u, the sequence of each user is decoded symbol-wise.

### Claim 11 (depends on 7)

The finite-field multiple access system of claim 7, wherein in the uplink transmission method using the finite-field resource, in a process that ŷ is constructed and then input into the decoder decoding restoration sequence {circumflex over (v)} of the channel encoding W, if soft decoding is used, a prior probability of the rn,i prior probability distribution is
Pr={CJ0/2J,CJ1/2J, . . . , CJJ−1/2J,CJJ/2J}.