Multichannel Conflict-Avoiding Codes for Expanded Scenarios

1 Multichannel Conflict-A v oiding Codes for Expanded Scenarios Kangkang Xu, Y uan-Hsun Lo, IEEE Member , Tsai-Lien W ong, Y ijin Zhang, IEEE Senior Member , K enneth W . Shum, IEEE Senior Member Abstract A conflict-av oiding code (CA C) of length L and weight w is used for deterministic multiple-access without feedback. When the number of simultaneous activ e users is less than or equal to w , such a code is able to provide a hard guarantee that each acti ve user has a successful transmission within e very consecutiv e L time slots. Recently , CA Cs were extended to multichannel CAcs (MC-CA Cs) ov er M > 1 orthogonal channels with the aim of increasing the number of potential users that can be supported. While most existing results on MC-CA C are deriv ed under the assumption M ≥ w , this paper focuses on the case M < w , which is more relev ant to practical application scenarios. In this paper, we first introduce the concept of exceptional codewords in MC-CA Cs. By employing Kneser’ s Theorem and some techniques from additi ve combinatorics, we deri ve a series of optimal MC-CACs. Along the way , sev eral pre viously kno wn optimal CA C results are generalized. Finally , our results e xtend naturally to AM-OPPTS MC-CA Cs and mixed-weight MC-CA Cs, two classes of rele vant codes. This work was supported in part by the National Science and T echnology Council of T aiwan under Grants 113-2115-M- 110-003-MY2 and 114-2628-M-153-001-MY3, and in part by the National Natural Science Foundation of China under Grant 62071236. K. Xu and Y .-H. Lo are with the Department of Applied Mathematics, National Pingtung University , T aiwan. Email: ivykkxu107@gmail.com, yhlo0830@gmail.com T .-L. W ong is with the Department of Applied Mathematics, National Sun Y at-sen Univ ersity , T aiwan. Email: tl- wong@math.nsysu.edu.tw Y . Zhang is with the School of Electronic and Optical Engineering, Nanjing University of Science and T echnology , Nanjing 210094, China. Email: yijin.zhang@gmail.com K. W . Shum is with the School of Science and Engineering, The Chinese Univ ersity of Hong K ong at Shenzhen, Shenzhen 518100, China. Email: wkshum@cuhk.edu.cn February 26, 2026 DRAFT 2 I . I N T R O D U C T I O N A. Single-Channel Conflict-A voiding Codes Let Z L ≜ { 0 , 1 , . . . , L − 1 } denote the ring of residue modulo L . For S ⊆ Z L , let d ∗ ( S ) ≜ { a − b ( mod L ) : a, b ∈ S, a  = b } denote the set of (nonzer o) differ ences of S . Definition 1. Let L and w be two positiv e integers with L ≥ w . A conflict-avoiding code (CAC) C of length L with weight w is a collection of w -subsets, called code words, of Z L such that d ∗ ( S ) ∩ d ∗ ( S ′ ) = ∅ ∀ S, S ′ ∈ C , S  = S ′ . (1) The equation (1) is called the disjoint-differ ence-set pr operty . Let CAC( L, w ) denote the class of all CA Cs of length L with weight w . The maximum size of a code in CA C( L, w ) is denoted by K ( L, w ) , i.e., K ( L, w ) ≜ max {|C | : C ∈ CA C( L, w ) } . A code C ∈ CA C( L, w ) is called optimal if its code size achie ves K ( L, w ) . A w -subset S ⊆ Z L is said to be equi-differ ence with generator g ∈ Z ∗ L if S is of the form { 0 , g , 2 g , . . . , ( w − 1) g } . A CA C is called equi-dif ference if it entirely consists of equi-difference code words. Let CAC e ( L, w ) ⊂ CA C( L, w ) denote the class of all equi-difference codes and K e ( L, w ) be the maximum size among CAC e ( L, w ) . Obviously , K e ( L, w ) ≤ K ( L, w ) . B. Multichannel Conflict-A voiding Codes In a multiple access network with M channels, a transmission pattern is represented as a zero-one M × L array , where M denotes the number of channels and L refers to the code word length. The ro ws and columns of the array are further index ed by I M ≜ { 1 , 2 , ..., M } and Z L , respecti vely . In this setting, a code word with weight w can be realized as a w -subset in I M × Z L , in which each element is an ordered pair . For two indices i, j ∈ I M , define the ( i, j ) set of differ ences by D S ( i, j ) :=      { t 1 − t 2 ( mod L ) : t 1 ∈ S i , t 2 ∈ S j } if i  = j, { t 1 − t 2 ( mod L ) : t 1 ∈ S i , t 2 ∈ S j } \ { 0 } if i = j. (2) DRAFT February 26, 2026 3 Note that, in the identical-index case, D S ( i, i ) = d ∗ ( S i ) . D S is called the array of differ ences of the transmission pattern S . T wo arrays of differences D S and D S ′ are said to be disjoint, denoted by D S ∩ D S ′ = ∅ , if they are entry-wise disjoint. Definition 2. Let M , L and w be positi ve integers. A multichannel conflict-avoiding code (MC- CA C) C of length L with weight w in M channels is a collection of w -subsets, also called codewor ds , of I M × Z L such that D S ∩ D S ′ = ∅ ∀ S, S ′ ∈ C , S  = S ′ . (3) The equation (3) is called the disjoint-differ ence-array pr operty . Let MC-CA C ( M , L, w ) denote the class of all MC-CA Cs of length L with weight w in M channels. The maximum size of a code in MC-CA C ( M , L, w ) is denoted by K ( M , L, w ) , i.e., K ( M , L, w ) ≜ max {|C | : C ∈ MC-CA C ( M , L, w ) } . A code C ∈ MC-CA C ( M , L, w ) is called optimal if its code size reaches K ( M , L, w ) . C. Contribution The technical results and main contributions of this paper are summarized below . 1) Define the so-called exceptional codew ords in MC-CA Cs, and deriv e a series of related properties by means of Kneser’ s theorem. See Lemma 2 and Lemma 3 in Section II. 2) Obtain two classes of optimal single-channel CA Cs of lengths p r 1 1 · · · p r n n and ( w − 1) p r 1 1 · · · p r n n , where p i are primes and w refers to the Hamming weight. See Theorems 7 and 10 in Section IV. The constructions of these codes can be extended to mixed-weight CA Cs. 3) Obtain optimal MC-CA Cs of length ( w − 1) p r 1 1 · · · p r n n with two channels. See Theorem 13 in Section V. The construction of these codes can be extended to mixed-weight MC-CA Cs, which are defined for the first time in this paper . 4) Propose a construction of MC-CA Cs of length (2 w M − 1) L ′ , where w and M refer to the Hamming weight and the number of channels. See Theorem 14 in Section VI. 5) Deriv e a general upper bound on the maximum size of an MC-CA C, where the code length is with some restriction. This can be seen as a generalization of some pre viously-known results. See Theorem 4 in Section III. 6) Some applications to MC-CA Cs with at most one-packet per time slot property . See The- orems 17, 18 and 19 in Section VII-A. February 26, 2026 DRAFT 4 I I . E X C E P T I O NA L C O D E W O R D S In a (single-channel) CA C in CA C( L, w ) , a code word S is called exceptional if | d ∗ ( S ) | < 2 w − 2 [1], [2], [5]. In this section, we will extend the concept of exceptional code words to MC-CA Cs, together with some characterizations of structural properties. A. Exceptional P airs and Revisit of the K enser’ s Theor em W e first giv e some notion of Additi ve Combinatorics [6]. For two subsets A, B ⊆ Z L and an element x ∈ Z L , define x + A ≜ { x + a : a ∈ A } , A + B ≜ { a + b : a ∈ A, b ∈ B } , and (4) A − B ≜ { a − b : a ∈ A, b ∈ B } . Moreov er , define d ( A ) ≜ A − A. Note that 0 ∈ d ( A ) and d ( A ) \ { 0 } = d ∗ ( A ) , the set of nonzero dif ferences of A . In this notion, the definition of exceptional codew ords S in a CA C can be represented by | S − S | ≤ 2 | S | − 2 . W e first extend this concept to a pair of subsets. Definition 3. A pair of two subsets { A, B } is called exceptional if | A − B | ≤ | A | + | B | − 2 . (5) Note that | A − B | = | B − A | , so there is no confusion in writing A − B or B − A in (5). Let T be a non-empty subset in Z L . The set of stabilizers of T in Z L is defined as H ( T ) ≜ { h ∈ Z L : h + T = T } . It is obvious that 0 ∈ H ( T ) and H ( T ) is a subgroup of Z L . So, | H ( T ) | di vides L by Lagrange’ s theorem. T is called periodic if H ( T ) is non-trivial, namely , H ( T )  = { 0 } . Here we list some well-kno wn properties (e.g., see [2], [5]) of the set of stabilizers. Proposition 1. Let T ⊆ Z L be non-empty . (i) H ( T ) is a subgr oup of Z L , and thus | H ( T ) | divides L . DRAFT February 26, 2026 5 (ii) If 0 ∈ T , then H ( T ) ⊆ T . (iii) If T is periodic, then T = S a ∈ T ( a + H ( T )) . Mor eover , | H ( T ) | divides | T | . The following lists some results about the number of stabilizers of the dif ference set of an exceptional subset, which can be seen in [5, Corollary 1 and Lemma 1]. Lemma 1 ( [5]) . Let A be a nonempty subset in Z L . If A is exceptional, then (i) 2 ≤ | H ( A − A ) | ≤ 2 | A | − 2 , (ii) | H ( A − A ) | does not divide 2 | A | − 1 , and (iii) | H ( A − A ) | does not divide | A | − 1 . The abov e results were based on Kneser’ s theorem, in which the formal description is giv en belo w . Theorem 1 ( [6], [7]) . Let A and B be two non-empty subsets in Z L , and let H = H ( A + B ) . Then, | A + B | ≥ | A + H | + | B + H | − | H | . (6) In particular , | A + B | ≥ | A | + | B | − | H | . (7) W e shall study the number of stabilizers of the difference set of an exceptional pair , which can be seen as generalizations of Lemma 1. Lemma 2. Let A, B be two nonempty subsets of Z L . If { A, B } is an exceptional pair , then (i) 2 ≤ | H ( A − B ) | ≤ | A | + | B | − 2 , and (ii) | H ( A − B ) | does not divide | A | + | B | − 1 . (iii) | H ( A − B ) | does not divide | A | − 1 when | A | = | B | . Pr oof. For notational con venience, denote by H = H ( A − B ) . (i) By (5) and (7), we hav e | A | + | B | − 2 ≥ | A − B | = | A + ( − B ) | ≥ | A | + | − B | − | H | = | A | + | B | − | H | , which implies that | H | ≥ 2 . Therefore, the set A − B is periodic. By Proposition 1(iii), we get | H | ≤ | A − B | , which is less than or equal to | A | + | B | − 2 because { A, B } is exceptional. February 26, 2026 DRAFT 6 (ii) Suppose to the contrary that | H | divides | A | + | B | − 1 . Since, by Proposition 1(i), H is a subgroup of Z L , we hav e H = − H , which implies that | − B + H | = | − ( B + H ) | = | B + H | . Moreov er , as B + H is the disjoint union of cosets of H , the size | B + H | is a multiple of | H | , and thus | B + H | ≥ | H | · ⌈| B | / | H |⌉ . Similarly , | A + H | ≥ | H | · ⌈| A | / | H |⌉ . By plugging A = A , B = − B into (6) yields that | A − B | ≥ | A + H | + | B + H | − | H | ≥ | H |  | A | | H |  +  | B | | H |  − 1  . Since { A, B } is an exceptional pair , namely , | A − B | ≤ | A | + | B | − 2 , it follo ws that | A | + | B | − 2 ≥ | H |  | A | | H |  +  | B | | H |  − 1  . (8) Let | A | = q 1 | H | + r 1 and | B | = q 2 | H | + r 2 , for some integers q 1 , q 2 ≥ 0 and 0 < r 1 , r 2 ≤ | H | . Note that 1 < r 1 + r 2 ≤ 2 | H | . (9) By (8), we hav e ( q 1 + q 2 ) | H | + r 1 + r 2 − 2 = | A | + | B | − 2 ≥ | H |  ( q 1 + 1) + ( q 2 + 1) − 1  , which implies that r 1 + r 2 − 2 ≥ | H | . (10) Combining (9) and (10) yields | H | + 1 ≤ r 1 + r 2 − 1 ≤ 2 | H | − 1 . (11) By the assumption that | H | divides | A | + | B | − 1 = | H | ( q 1 + q 2 ) + r 1 + r 2 − 1 , we get | H | di vides r 1 + r 2 − 1 . This contradicts to (11). (iii) This case can be dealt with in the same argument we provided for (ii). By letting | A | = | B | = q | H | + r , i.e., q = q 1 = q 2 and r = r 1 = r 2 , the two inequalities (9) and (10) imply | H | 2 ≤ r − 1 ≤ | H | − 1 . (12) Suppose to the contrary that | H | divides | A | − 1 = q | H | + r − 1 . W e get | H | di vides r − 1 , which is a contradiction to (12). Hence the proof is completed. DRAFT February 26, 2026 7 B. Exceptional Codewor ds in MC-CA Cs Consider a subset S ∈ I M × Z L . For i ∈ I M , define S i ≜ { t : ( i, t ) ∈ S } . Obviously , | S | = P i ∈I M | S i | . W e further define e S ≜ |{ i ∈ I M : S i  = ∅}| indicating the number of channels that hav e at least one packet in the transmission pattern S . W ith the notation giv en in (4), the ( i, j ) set of difference of S , D S ( i, j ) , defined in (2) can be re written as S i − S j if i  = j . W e are ready to define exceptional subsets in I M × Z L . Definition 4. A subset S ∈ I M × Z L is called exceptional if at least one of the follo wings holds. (i) There exists i ∈ I M such that S i is exceptional, i.e., | S i − S i | ≤ 2 | S i | − 2 . (ii) There exist distinct indices i, j ∈ I M such that { S i , S j } is exceptional, i.e., | S i − S j | ≤ | S i | + | S j | − 2 . Lemma 3. Let S be a w -subset in I M × Z L . If the prime factors of L ar e all lar ger than or equal to 2 w − 1 , then S is not exceptional. Pr oof. Suppose to the contrary that S is exceptional. Let T be an y nonempty subset in Z L . Since all prime factors L are all larger than or equal to 2 w − 1 , by Proposition 1(i), we hav e | H ( T ) | = 1 or | H ( T ) | ≥ 2 w − 1 . (13) If there exists an index i ∈ I M such that S i is exceptional, by Lemma 1(i), we have 2 ≤ | H ( S i − S i ) | ≤ 2 | S i | − 2 ≤ 2 w − 2 . If there exist distinct indices i, j ∈ I M such that { S i , S j } is exceptional, by Lemma 2(i), we hav e 2 ≤ | H ( S i − S j ) | ≤ | S i | + | S j | − 2 ≤ w − 2 . Either case contradicts to (13), so the proof is done. W e end this section with a fundamental result in Group Theory , which will be used in subsequent sections. Proposition 2. The subgroup of Z L is uniquely determined by its order . More precisely , for any di visor d of L , the unique subgroup of Z L with order d is { 0 , L/d, 2 L/d, . . . , ( d − 1) L/d } . February 26, 2026 DRAFT 8 I I I . G E N E R A L U P P E R B O U N D S O N T H E M A X I M U M C O D E S I Z E W e first recall an upper bound on the maximum code size of a CA C with some restriction on the code length. Theorem 2 ( [1], Theorem 5) . Let L and w be positive inte gers. If the prime factors of L ar e all lar ger than or equal to 2 w − 1 , then K ( L, w ) ≤ L − 1 2 w − 2 . The follo wing result is due to [4]. Theorem 3 ( [4], Corollary 13) . Let M , L and w be positive inte gers. If the prime factor s of L ar e all lar ger than or equal to 2 w − 1 , we have K ( M , L, w ) ≤ M ( M − 1) L w ( w − 1) + M ( L − 1) 2 w − 2 . (14) The upper bounds in Theorem 3 can be improv ed in the case when M < w , which is shown in the follo wing theorem. Theorem 4. Let M , L and w be positive inte gers with M < w . If the prime factors of L ar e all lar ger than or equal to 2 w − 1 , we have K ( M , L, w ) ≤ M ( M − 1) L (2 w − M )( w − 1) + M ( L − 1) 2 w − 2 . (15) Pr oof. Let C be any code in MC-CA C ( M , L, w ) . The codew ords in C are partitioned, according to the number of channels that are occupied in their transmission patterns, i.e., e S , into the follo wing three classes: C 1 = { S ∈ C : e S = 1 } , C 2 = { S ∈ C : e S = M } , and C 3 = { S ∈ C : 2 ≤ e S ≤ M − 1 } . Since the prime factors of L are all larger than or equal to 2 w − 1 , by Lemma 3, each code word in C is non-exceptional. Then, we have | S i − S j | ≥ | S i | + | S j | − 1 for any i, j ∈ I M , and in particular , | d ∗ ( S i ) | = | S i − S i | − 1 ≥ 2( | S i | − 1) . DRAFT February 26, 2026 9 For any i ∈ I M , by the disjoint-dif ference-array property , we have L − 1 = | Z ∗ L | ≥ X S ∈C | D S ( i, i ) | . Hence, M ( L − 1) ≥ X i ∈I M X S ∈C | D S ( i, i ) | = X S ∈C X i ∈I M ,S i  = ∅ | d ∗ ( S i ) | ≥ X S ∈C X i ∈I M ,S i  = ∅ 2( | S i | − 1) = 2( w − 1) |C 1 | + 2( w − M ) |C 2 | + 2 X S ∈C 3 ( w − e S ) , which implies that ( w − 1) |C 1 | + ( w − M ) |C 2 | + X S ∈C 3 ( w − e S ) ≤ M ( L − 1) 2 . (16) No w we consider D S ( i, j ) for distinct indices i, j ∈ I M , by going through all code words in C . The sets of dif ferences D S ( i, j ) for S ∈ C are mutually disjoint, and their union is a subset of Z L . As D S ( i, j ) = ∅ for S ∈ C 1 , we hav e L = | Z L | ≥ X S ∈C 2 ∪C 3 | D S ( i, j ) | . (17) Therefore, M ( M − 1) L ≥ X i,j ∈I M ,i  = j X S ∈C 2 ∪C 3 | D S ( i, j ) | ≥ X S ∈C 2 ∪C 3 X i,j ∈I M ,i  = j S i ,S j  = ∅ ( | S i | + | S j | − 1) = X S ∈C 2 ∪C 3   2( e S − 1)   X i ∈I M ,S i  = ∅ | S i |   − e S ( e S − 1)   = X S ∈C 2 ∪C 3 ( e S − 1)(2 w − e S ) ≥ ( M − 1)(2 w − M ) |C 2 | + X S ∈C 3 ( e S − 1)(2 w − e S ) , yields that ( M − 1)(2 w − M ) |C 2 | + X S ∈C 3 ( e S − 1)(2 w − e S ) ≤ M ( M − 1) L. (18) February 26, 2026 DRAFT 10 W e multiply the inequality in (16) by (2 w − M ) and combine the result with (18). W e then get ( w − 1)(2 w − M ) ( |C 1 | + |C 2 | ) + X S ∈C 3  − e 2 S + ( M + 1) e S + 2 w 2 − M w − 2 w  ≤ M ( L − 1) 2 (2 w − M ) + M ( M − 1) L. (19) When M = 2 , we hav e C 3 = ∅ , and thus it follows from (19) that |C | = |C 1 | + |C 2 | ≤ M ( M − 1) L (2 w − M )( w − 1) + M ( L − 1) 2 w − 2 , as desired. So, we assume M ≥ 3 in what follows. Define a function f ( x ) = − x 2 + ( M + 1) x + 2 w 2 − M w − 2 w . The deri vati ve of f ( x ) is − 2 x + M + 1 , implying that f ( x ) is a concav e function with a unique maximum at x = M +1 2 . Let us focus on the interv al [2 , M − 1] . Since M +1 2 ∈ [2 , M − 1] for M ≥ 3 and f (2) = (2 w − M )( w − 1) + M − 2 = f ( M − 1) , we hav e f ( x ) ≥ min { f (2) , f ( M − 1) } = (2 w − M )( w − 1) + M − 2 , for 2 ≤ x ≤ M − 1 . Therefore, f ( e S ) ≥ (2 w − M )( w − 1) + M − 2 ≥ (2 w − M )( w − 1) (20) since 2 ≤ e S ≤ M − 1 and M ≥ 3 for S ∈ C 3 . Therefore, the left-hand-side of (19) turns to ( w − 1)(2 w − M ) ( |C 1 | + |C 2 | ) + X S ∈C 3 f ( e S ) ≥ ( w − 1)(2 w − M ) ( |C 1 | + |C 2 | + |C 3 | ) (21) = ( w − 1)(2 w − M ) |C | . Then the inequality in (15) follo ws. Remark 1. If M ≥ 3 , the equality (15) will hold only when C 3 = ∅ , that is, all in volv ed code words S has either e S = 1 or e S = M . This fact can be seen from (20) – (21): If M ≥ 3 , the inequality in (20) turns out to be f ( e S ) > (2 w − M )( w − 1) , which leads to a strict inequality in (21) whene ver C 3  = ∅ . The upper bounds stated in Theorem 3 and Theorem 4 remain v alid when M = 1 . So, both they can be viewed as generalizations of Theorem 2. DRAFT February 26, 2026 11 It is worth mentioning that the proof method used for Theorem 4 is also applicable to Theorem 3 b ut it differs from the original proof gi ven in [4]. For completeness, we provide the new proof of Theorem 3 in Appendix A. Moreover , our proof in Appendix A can further re veal the fact that the equality in (14) will hold only when the code words S with either e S = 1 or e S = w are considered. Note that all the optimal MC-CA Cs obtained in [3], [4] hav e this property . I V . S I N G L E - C H A N N E L C A C S W e first introduce the p -ary representation of a positiv e integer . gi ven a positiv e integer n , let Z × n ≜ { x ∈ Z n : gcd( x, n ) = 1 } . Z × n is the set of units (i.e., in vertible elements) in Z ∗ n , and thus is a multiplicativ e group. Note that Z × n = Z ∗ n when n is a prime. Let p be an odd prime and r a positi ve inte ger . For c ∈ Z p r , consider the p -ary representation c = c 0 + c 1 p + · · · + c r − 1 p r − 1 . For t = 0 , 1 , . . . , r − 1 , let L t be the collection of c ∈ Z ∗ p r whose nonzero least significant digit in its p -ary representation is p t . Obviously , | L t | = ( p − 1) p r − t − 1 , and L 0 , L 1 , . . . , L r − 1 form a partition of Z ∗ p r , i.e., Z ∗ p r = L 0 ⊎ L 1 ⊎ · · · ⊎ L r − 1 . Integers in L t are called in the t -th layer . For a non-empty A ⊆ Z ∗ p , we arise it to a subset in Z ∗ p r , for any positiv e integer r , by defining S r ( A ) ≜ A 0 ⊎ A 1 ⊎ · · · ⊎ A r − 1 , (22) where A t = { c ∈ L t : c t ∈ A } . S r ( A ) is the collection of elements in Z ∗ p r whose nonzero least significant digit values in their p -ary representation are in A . Obviously , | A t | = | A | p r − 1 − t for each t , and thus |S r ( A ) | = | A |  1 + p + · · · + p r − 1  = | A | p r − 1 p − 1 . (23) The following is a useful property of the p -ary representation of c ∈ Z ∗ p r , which can also be found in [5, Proposition 3]. Proposition 3 ([5]) . Let p be an odd prime and r be a positive inte ger . F or j ∈ L 0 and c ∈ L t , 0 ≤ t ≤ r − 1 , one has j c ∈ L t and ( j c ) t = j 0 · c t ( mo d p ) . (24) Note that L 0 = Z × p r , which is the set of units in Z ∗ p r . February 26, 2026 DRAFT 12 A. Codewor d Length L = ( w − 1) p r 1 1 p r 2 2 · · · p r n n Gi ven a prime p , an element a ∈ Z ∗ p is called a quadratic r esidue if there exists an integer x ∈ Z ∗ p such that a = x 2 ; otherwise, a is called a quadratic non-r esidue . For a ∈ Z ∗ p , denote by  a p  = 1 if a is a quadratic residue modulo p and − 1 otherwise. The notation  a p  is called the Le gendr e symbol . It can be shown (e.g., [8]) that the Legendre symbol is multiplicati ve:  ab p  =  a p   b p  . (25) This subsection is dev oted to generalizing the following result, which is due to [5, Theorem 6]. Theorem 5 ([5]) . Let w be a positive inte ger and p be a prime such that p ≥ w . If  − 1 p  = − 1 and  j p   j − w + 1 p  = − 1 , ∀ j = 1 , 2 , . . . , w − 2 , then for any inte ger r ≥ 1 , K (( w − 1) p r , w ) = p r − 1 2 . W e first introduce the following construction. Theorem 6. Let w be a positive inte ger , and let p 1 < p 2 < · · · < p n be primes satisfying p 1 ≥ w . If, for each 1 ≤ i ≤ n , it holds that  − 1 p i  = − 1 (26) and  j p i   j − w + 1 p i  = − 1 , ∀ j = 1 , 2 , . . . , w − 2 , (27) then for any positive inte gers r 1 , . . . , r n , ther e exists a code C ∈ CA C e ( L, w ) with |C | = p r 1 1 · · · p r n n − 1 2 , wher e L = ( w − 1) p r 1 1 · · · p r n n . Pr oof. For conv enience, let L ′ = p r 1 1 · · · p r n n . Then, L = ( w − 1) L ′ . Since w − 1 , p 1 , p 2 , . . . , p n are pairwise coprime, one has Z L ∼ = Z w − 1 × Z L ′ and Z L ′ ∼ = Z p r 1 1 × · · · × Z p r n n . DRAFT February 26, 2026 13 In what follows, we use ordered pairs ( z , x ) ∈ Z w − 1 × Z L ′ to represent elements in Z L , where x ∈ Z L ′ are further represented by n -tuples ( x 1 , . . . , x n ) ∈ Z p r 1 1 × · · · × Z p r n n . Let Q i be the set of quadratic residues modulo p i . For i = 1 , 2 , . . . , n define b Q i ≜ { (0 , . . . , 0 , g , x i +1 , . . . , x n ) ∈ Z p r 1 1 × · · · × Z p r n n : g ∈ S r i ( Q i ) , x t ∈ Z p r t t for i < t ≤ n } . Let b Q ≜ n ] i =1 b Q i (28) be the disjoint union of b Q 1 , b Q 2 , . . . , b Q n . As | Q i | = ( p i − 1) / 2 , by (23), we have | b Q | = n X i =1 p r i i − 1 2 n Y j = i +1 p r j j = p r 1 1 · · · p r n n − 1 2 . For a ∈ b Q , define S a = { j (1 , a ) ∈ Z w − 1 × Z L ′ : j = 0 , 1 , 2 . . . , w − 1 } be the w -subset generated by a whose set of nonzero dif ferences is of the form d ∗ ( S a ) = {± j (1 , a ) ∈ Z w − 1 × Z L ′ : j = 1 , 2 , . . . , w − 1 } (29) = {± j (1 , a ) ∈ Z w − 1 × Z L ′ : j = 1 , 2 , . . . , w − 2 } ∪ { (0 , ± ( w − 1) a ) } . (30) Assume a = (0 , . . . , 0 , g , x i +1 , . . . , x n ) ∈ Z r 1 p 1 × · · · × Z r n p n , i.e., a ∈ b Q i , then the elements in d ∗ ( S a ) are of the form ± j (1 , a ) = ± j (1 , (0 , . . . , 0 , g , x i +1 , . . . , x n )) or (0 , ± ( w − 1) a ) = (0 , (0 , . . . , 0 , ± ( w − 1) g , ± ( w − 1) x i +1 , . . . , ± ( w − 1) x n )) . In what follows, we shall prov e that b Q can generate the desired code C , that is, d ∗ ( S a ) , a ∈ b Q are mutually disjoint. Assume a ∈ b Q i and b ∈ b Q i ′ , it is obvious that d ∗ ( S a ) ∩ d ∗ ( S b ) = ∅ whenev er i  = i ′ . So, it suffices to consider a and b are in the same set b Q i , for some i . Suppose to the contrary that d ∗ ( S a ) ∩ d ∗ ( S b )  = ∅ , where a = (0 , . . . , 0 , g , x i +1 , . . . , x n ) and b = (0 , . . . , 0 , h, y i +1 , . . . , y n ) for some g , h ∈ S r i ( Q i ) . W ithout loss of generality generality , February 26, 2026 DRAFT 14 assume j (1 , (0 , . . . , 0 , g , x i +1 , . . . , x n )) is one of the common elements in d ∗ ( S a ) ∩ d ∗ ( S b ) , where 1 ≤ j ≤ w − 1 . Then we ha ve j (1 , (0 , . . . , 0 , g , x i +1 , . . . , x n )) = ± k (1 , (0 , . . . , 0 , h, y i +1 , . . . , y n )) (31) for some k ∈ { 1 , 2 , . . . , w − 1 } . There are two cases. Case 1: j = w − 1 . From (31) we hav e ( w − 1) g = ± ( w − 1) h , which implies g = − h (mod p r i i ) due to g  = h . By considering the p i -ary representation, both g and − h are in the same layer , say L t for some t , and thus g t = ( − h ) t (mod p i ). Note ( − h ) t = ( − 1) · h t (mod p i ) by Proposition 3. By (25), we hav e  g t p i  =  ( − h ) t p i  =  − 1 p i   h t p i  . Ho wev er , g t and h t are quadratic residues modulo p i by the assumption that g , h ∈ S r i ( Q i ) and the fact that g , h ∈ L t . It further implies that  − 1 p i  = 1 , which contradicts the condition in (26). Case 2: 1 ≤ j ≤ w − 2 . From (31) we have j (1 , g ) = ± k (1 , h ) in Z w − 1 × Z p r i i , where 1 ≤ k ≤ w − 2 . If j (1 , g ) = k (1 , h ) , it follows that g = h , where a contradiction occurs. If j (1 , g ) = − k (1 , h ) , we ha ve j + k = w − 1 , and then get j g = ( j − w + 1) h (mod p r i i ). By considering the p i -ary representation, both j g and ( j − w + 1) h are in the same layer , say L t for some t . Since j ≤ w − 2 < p i , we ha ve j = j 0 and ( j − w + 1) = ( j − w + 1) 0 (mod p i ), namely , both j and j − w + 1 are in L 0 . It follo ws from Proposition 3 that g , h ∈ L t . More precisely , g = j − 1 · j g ∈ L t and h = ( j − w + 1) − 1 · ( j − w + 1) h ∈ L t . Therefore, g t , h t ∈ Q i by assumption. By Eq. (25) and Proposition 3, ( j g ) t = (( j − w + 1) h ) t ( mo d p i ) ⇒ j · g t = ( j − w + 1) · h t ( mo d p i ) ⇒  j p i   g t p i  =  j − w + 1 p i   h t p i  ⇒  j p i  =  j − w + 1 p i  , which contradicts the condition in (27), and the proof is completed. Remark 2. Let C be the CA C constructed in Theorem 6. By (30), we have | d ∗ ( S ) | = 2( w − 1) for all S ∈ C . So, it costs ( w − 1)( L ′ − 1) distinct dif ferences in total, where L ′ = p r 1 1 · · · p r n n . DRAFT February 26, 2026 15 Moreov er , since p 1 ≥ w , it also rev eals from (30) that ( z , 0) / ∈ d ∗ ( S ) for all z ∈ Z w − 1 and S ∈ C . This concludes that [ S ∈C d ∗ ( S ) = Z w − 1 × Z ∗ L ′ . Theorem 7. Let w be a positive inte ger , and let p 1 < p 2 < · · · < p n be primes satisfying p 1 ≥ w . If the two conditions in (26) and (27) hold, then for any positive inte gers r 1 , . . . , r n , we have K (( w − 1) p r 1 1 · · · p r n n , w ) = p r 1 1 · · · p r n n − 1 2 pr ovided that t X i =1 (2 w − 1 − p i ) ≤ w − 1 , (32) wher e p t < 2 w − 1 ≤ p t +1 for some 0 ≤ t ≤ n . Note that p 0 ≜ 1 and p n +1 ≜ ∞ by con vention. Note also that (32) always holds for t = 0 , 1 . Pr oof. Let L ′ = p r 1 1 · · · p r n n . By Theorem 6, it suffices to sho w that for any code C ∈ CA C(( w − 1) L ′ , w ) , one has |C | ≤ L ′ − 1 2 . Let E ⊆ C be the collection of all exceptional code words in C . For notational con venience, denote by H S = H ( d ( S )) for S ∈ E . Pick any S ∈ E . Since H S is a subgroup of Z ( w − 1) L ′ , the size | H S | must divide ( w − 1) L ′ . By Lemma 1(iii), | H S | does not di vide ( w − 1) . As | H S | ≥ 2 by Lemma 1(i), then | H S | is a multiple of p i for some i . W e consider two cases. Case 1: p 1 > 2 w − 2 . In this case we hav e | H S | ≤ 2 w − 2 < p 1 due to Lemma 1(i). This is a contradiction to | H S | a multiple of some prime factor p i . In other words, there is no e xceptional code word in this case. Therefore, | d ∗ ( S ) | ≤ 2 w − 2 for all S ∈ C . By the disjoint-difference-set property , ( w − 1) L ′ − 1 = | Z ∗ ( w − 1) L ′ | ≥ [ S ∈C | d ∗ ( S ) | ≥ (2 w − 2) |C | , which implies that |C | ≤  L ′ − 1 2 + w − 2 2 w − 2  = L ′ − 1 2 . Case 2: p 1 ≤ 2 w − 2 . Since 0 ∈ d ( S ) for S ∈ E , if follows from Proposition 1(ii) that H S ⊆ d ( S ) . Then, H S ∩ H S ′ = { 0 } for any two distinct S, S ′ ∈ E . Moreov er , since H S is uniquely determined by its order due to Proposition 2, we have gcd( | H S | , | H S ′ | ) = 1 . Now , assume p t < 2 w − 1 ≤ p t +1 for some 1 ≤ t ≤ n . Since | H S | ≤ 2 w − 2 < p t +1 , for S ∈ E , February 26, 2026 DRAFT 16 the prime factors of | H S | are in { p 1 , . . . , p t } . This concludes that |E | ≤ t and | H S | , S ∈ E , are pairwise coprime. Therefore, X S ∈E (2 w − 1 − | H S | ) ≤ t X i =1 (2 w − 1 − p i ) . (33) On the other hand, by the disjoint-dif ference-set property , ( w − 1) L ′ − 1 = | Z ∗ ( w − 1) L ′ | ≥ [ S ∈C | d ∗ ( S ) | ≥ (2 w − 2)( |C | − |E | ) + X S ∈E | d ∗ ( S ) | . Since | H S | ≤ | d ( S ) | = | d ∗ ( S ) | + 1 due to Proposition 1(ii), it follows that ( w − 1) L ′ − 1 ≥ (2 w − 2) |C | − X S ∈E (2 w − 1 − | H S | ) . (34) Combining (32) – (34) yields |C | ≤  ( w − 1) L ′ + w − 2 2 w − 2  = L ′ − 1 2 +  2 w − 3 2 w − 2  = L ′ − 1 2 , as desired. The primes that satisfy the two conditions (26) and (27) are characterized in [5] with the help of Gauss’ s lemma and laws of quadratic reciprocity , which lead to a series of optimal CA Cs of length ( w − 1) p r in [5, Corollary 2]. As an application of Theorem 7, the following corollary generalizes these optimal CA Cs. Corollary 1. Let L ′ = Q n i =1 p r i i , wher e p i ar e distinct primes and r i ar e any positive inte gers. One has 1) K (3 L ′ , 4) = ( L ′ − 1) / 2 if p i ≡ ± 1 ( mod 8) for all i , 2) K (4 L ′ , 5) = ( L ′ − 1) / 2 if p i ≡ − 1 ( mod 12) for all i , 3) K (5 L ′ , 6) = ( L ′ − 1) / 2 if p i ≡ − 1 , − 5 ( mo d 24) for all i , 4) K (6 L ′ , 7) = ( L ′ − 1) / 2 if p i ≡ − 1 , − 9 ( mo d 40) for all i , 5) K (7 L ′ , 8) = ( L ′ − 1) / 2 if p i ≡ − 1 , − 49 ( mo d 120) for all i , 6) K (8 L ′ , 9) = ( L ′ − 1) / 2 if p i ≡ − 1 , 59 , − 109 , − 121 , 131 , − 169 ( mo d 420) for all i , 7) K (9 L ′ , 10) = ( L ′ − 1) / 2 if p i ≡ − 1 , − 9 , 31 , − 81 , 111 , − 121 ( mo d 280) for all i , and 8) K (10 L ′ , 11) = ( L ′ − 1) / 2 if p i ≡ − 1 , − 5 , − 25 , 43 , 47 , 67 ( mo d 168) for all i . Note that we can deri ve a sufficient condition of primes p i so that K (( w − 1) L ′ , w ) = ( L ′ − 1) / 2 , for any arbitrary w . Please refer to [5] for more details. DRAFT February 26, 2026 17 B. Codewor d Length L = p r 1 1 p r 2 2 · · · p r n n This subsection is dev oted to generalize the following result, which is due to [5, Theorem 10]. Theorem 8 ([5]) . Let w be a positive inte ger , and let p be a prime such that p − 1 is divided by 2 w − 2 . If ther e is a code in CA C e ( p, w ) with ( p − 1) / (2 w − 2) codewor ds, then for any integ er r ≥ 1 , K ( p r , w ) = p r − 1 2 w − 2 . W e first introduce the following construction. Theorem 9. Let w be a positive inte ger , and let p 1 < p 2 < · · · < p n be primes satisfying p 1 ≥ 2 w − 1 . If ther e is a code in CA C e ( p i , w ) with m i codewor ds for each i , then for any positive inte gers r 1 , . . . , r n , ther e exists a code C ∈ CA C e ( L, w ) with |C | = n X i =1 m i ( p r i i − 1) p i − 1 n Y j = i +1 p r j j , (35) wher e L = p r 1 1 · · · p r n n . Pr oof. Since p 1 , p 2 , . . . , p n are distinct primes, Z L ∼ = Z p r 1 1 × Z p r 2 2 × · · · × Z p r n n . In what follows, we use n -tuples ( x 1 , . . . , x n ) ∈ Z p r 1 1 × · · · × Z p r n n to represent elements in Z L . For i = 1 , 2 , . . . , n , let Γ i be a set of m i generators of the giv en code in CAC e ( p i , w ) . Then, define b Γ i ≜ { (0 , . . . , 0 , g , x i +1 , . . . , x n ) ∈ Z p r 1 1 × · · · × Z p r n n : g ∈ S r i (Γ i ) , x i ∈ Z r i p i for i < j ≤ n } . It is easy to see from (23) that for i = 1 , 2 , . . . , n , | b Γ i | = m i ( p r i i − 1) p i − 1 n Y j = i +1 p r j j . Note that the sets b Γ 1 , b Γ 2 , . . . , b Γ n are mutually disjoint. Let b Γ = n ] i =1 b Γ i (36) February 26, 2026 DRAFT 18 be the disjoint union of these sets. Obviously , | b Γ | = n X i =1 m i ( p r i i − 1) p i − 1 n Y j = i +1 p r j j . Consider the set b Γ . For a ∈ b Γ , if a = (0 , . . . , 0 , g , x i +1 , . . . , x n ) for some g ∈ S r i (Γ i ) , i.e., a ∈ b Γ i , then define S a = { j a ∈ Z p r 1 1 × · · · × Z p r n n : j = 0 , 1 , 2 , . . . , w − 1 } (37) be the w -subset generated by a whose set of nonzero dif ferences is of the form d ∗ ( S a ) = { j a ∈ Z p r 1 1 × · · · × Z p r n n : j = ± 1 , ± 2 , . . . , ± ( w − 1) } . (38) In what follows, we shall prove that b Γ is the set of generators of the desired code C , that is, d ∗ ( S a ) ∩ d ∗ ( S b ) = ∅ for any distinct a, b ∈ b Γ . It suffices to consider that a and b are in the same subset b Γ i , for some i , because the assertion is definitely true in the other case. Suppose to the contrary that d ∗ ( S a ) ∩ d ∗ ( S b )  = ∅ , where a = (0 , . . . , 0 , g , x i +1 , . . . , x n ) and b = (0 , . . . , 0 , h, y i +1 , . . . , y n ) for some g , h ∈ S r i (Γ i ) . Assume j (0 , . . . , 0 , g , x i +1 , . . . , x n ) = k (0 , . . . , 0 , h, y i +1 , . . . , y n ) for some j, k ∈ {± 1 , ± 2 , . . . , ± ( w i − 1) } . If j = k , then g = h and x t = y t for i + 1 ≤ t ≤ n , namely , a = b . So, we consider j  = k in what follo ws. Let us focus on the i -th component, namely , j g = k h . Since p i ≥ 2 w − 1 , by Proposition 3, one has j g ∈ L t for every j ∈ {± 1 , ± 2 , . . . , ± ( w − 1) } if g ∈ L t . Here we use L t to denote the t -th layer in the p i -ary representation of elements of Z p r i i for notatinoal con venience. It follo ws that j g  = k h for any j, k ∈ {± 1 , ± 2 , . . . , ± ( w − 1) } whene ver g and h are in distinct layers. Therefore, we only need to consider the case when both g and h are in the same layer , say L t for some t . Observe that j, k ∈ L 0 due to p i ≥ 2 w − 1 . By Proposition 3 again, j · g t = k · h t (mod p i ). If g t  = h t , by the assumption that g t , h t ∈ Γ i be two distinct generators in the gi ven code in CA C e ( p i , w ) , one has j · g t  = k · h t . If g t = h t , it further implies that ( j − k ) g t = 0 (mod p t ), which is impossible because of j, k beging distinct in {± 1 , ± 2 , · · · , ± ( w − 1) } and p i ≥ 2 w − 1 . This completes the proof. Theorem 10. Let w be a positive inte ger , and let p 1 < · · · < p n be primes such that p i − 1 is divided by 2 w − 2 for each i . If ther e is a code in CA C e ( p i , w ) with ( p i − 1) / (2 w − 2) codewor ds for each i , then for any positive inte gers r 1 , . . . , r n , K ( p r 1 1 · · · p r n t , w ) = p r 1 1 · · · p r t t − 1 2( w − 1) . (39) DRAFT February 26, 2026 19 Pr oof. The assumption that p i − 1 is divided by 2 w − 2 guarantees p i ≥ 2 w − 1 , for each i . By plugging m i = ( p i − 1) / (2 w − 2) into (35), there exists a code in CAC e ( p r 1 1 · · · p r n n , w ) with ( p r 1 1 · · · p r n n − 1) / (2 w − 2) codew ords. Since the least prime factor p 1 is larger than or equal to 2 w − 1 , the result follows by Theorem 2. Remark 3. Let C be an optimal CA C considered in Theorem 10. Since p 1 ≥ 2 w − 1 , it follows from (38) that | d ∗ ( S ) | = 2( w − 1) for S ∈ C . As |C | = ( L − 1) / 2 , where L = p r 1 1 · · · p r n n , we hav e [ S ∈C d ∗ ( S ) = Z ∗ L . So, the optimal CA Cs obtained in Theorem 9 is tight [9]. V . M U LT I C H A N N E L C A C S W I T H T W O C H A N N E L S Theorem 11. Let w be a positive inte ger , and let p 1 < p 2 < · · · < p n be primes satisfying p 1 ≥ w . F or each 1 ≤ i ≤ n , if there is a code in CAC e ( p i , w ) with m i codewor ds and the two conditions in (26) and (27) hold, then for any positive inte gers r 1 , . . . , r n , ther e e xists a code C ∈ MC-CA C (2 , L, w ) with |C | = n Y i =1 p r i i + n X i =1 2 m i ( p r i i − 1) p i − 1 n Y j = i +1 p r j j (40) codewor ds, wher e L = ( w − 1) p r 1 1 p r 2 2 · · · p r n n . Pr oof. Let L ′ = p r 1 1 p r 2 2 · · · p r n n , then L = ( w − 1) L ′ . In the follo wing construction, the desired code C will consist of three classes of code words C 1 , C 2 and C 3 with sizes |C 1 | = L ′ − 1 , |C 2 | = n X i =1 2 m i ( p r i i − 1) p i − 1 n Y j = i +1 p r j j , and |C 3 | = 1 , where the code words in C 1 and C 2 are from the constructions in Theorem 6 and Theorem 9, respecti vely . Since gcd( w − 1 , L ′ ) = 1 , we have Z ∼ = Z w − 1 × Z L ′ . Elements in I 2 × Z L can be represented by ( m, ( z , x )) , where m ∈ I 2 = { 1 , 2 } and z ∈ Z w − 1 , x ∈ Z L ′ . Since the two conditions (26) – (27) hold, Theorem 6 ensures the existence of a code in CA C( L, w ) with ( L ′ − 1) / 2 codew ords. W e adopt the notation in the proof of Theorem 6. February 26, 2026 DRAFT 20 Consider the set b Q defined in (28). Note that b Q ⊂ Z L ′ with size | b Q | = ( L ′ − 1) / 2 . F or a ∈ b Q define the follo wing two codew ords in I 2 × Z L S 1 a = { (1 , (1 , − a )) } ∪ { (2 , j (1 , a )) : j = 0 , 1 , 2 , . . . , w − 2 } , and S 2 a = { (1 , j (1 , a )) : j = 0 , 1 , 2 , . . . , w − 2 } ∪ { (2 , (1 , − a )) } . Let C 1 = { S 1 a , S 2 a : a ∈ b Q } be the collection of these code words deri ved from all elements in b Q . Let us focus on the arrays of dif ferences D S 1 a and D S 2 a . It is easy to see that D S 1 a (1 , 1) = D S 2 a (2 , 2) = ∅ , (41) D S 1 a (2 , 2) = D S 2 a (1 , 1) = {± j (1 , a ) ∈ Z w − 1 × Z L ′ : j = 1 , 2 , . . . , w − 2 } , (42) D S 1 a (1 , 2) = D S 2 a (2 , 1) = {− j (1 , a ) ∈ Z w − 1 × Z L ′ : j = 1 , 2 , . . . , w − 1 } , (43) D S 1 a (2 , 1) = D S 2 a (1 , 2) = { j (1 , a ) ∈ Z w − 1 × Z L ′ : j = 1 , 2 , . . . , w − 1 } . (44) Recall that in the proof of Theorem 6, it was sho wn that d ∗ ( S a ) ∩ d ∗ ( S b ) = ∅ for any two distinct a, b ∈ b Q , where d ∗ ( S a ) is gi ven in (29) as d ∗ ( S a ) = {± j (1 , a ) ∈ Z w − 1 × Z L ′ : j = 1 , 2 , . . . , w − 1 } . Therefore, it follows from (41) – (44) that D S 1 a , D S 2 a are mutually disjoint ov er all a ∈ b Q . Moreov er , by the property giv en in Remark 2, we further deriv e that [ S ∈C 1 D S = Z ∗ w − 1 × Z ∗ L ′ Z w − 1 × Z ∗ L ′ Z w − 1 × Z ∗ L ′ Z ∗ w − 1 × Z ∗ L ′ . (45) Since there is a code in CA C e ( p i , w ) with m i code words for each i , it follows from Theorem 9 that there exists a code in CAC( L ′ , w ) with P n i =1  m i ( p r i i − 1) p i − 1 Q n j = i +1 p r j j  code words, where the generators are collected in the set b Γ gi ven in (36). W e adopt the notation in the proof of Theorem 9. For a ∈ b Γ define the follo wing two codew ords in I 2 × Z L ′ T 1 a = { (1 , (0 , j a )) : j = 0 , 1 , 2 , . . . , w − 1 } , and T 2 a = { (2 , (0 , j a )) : j = 0 , 1 , 2 , . . . , w − 1 } . Let C 2 = { T 1 a , T 2 a : a ∈ b Γ } be the collection of these code words. Consider the arrays of dif ferences D T 1 a and D T 2 a . It is obvious that D T 1 a (1 , 2) = D T 1 a (2 , 1) = D T 1 a (2 , 2) = D T 2 a (1 , 2) = D T 2 a (2 , 1) = D T 2 a (1 , 1) = ∅ , and (46) D T 1 a (1 , 1) = D T 2 a (2 , 2) = { (0 , ± j a ) ∈ Z w − 1 × Z L ′ : j = 1 , 2 , . . . , w − 1 } . (47) DRAFT February 26, 2026 21 In the proof of Theorem 9, it was sho wn that {± j a ∈ Z L ′ : j = 1 , 2 , . . . , w − 1 } , a ∈ b Γ , are mutually disjoint. Then, it follo ws from (46) – (47) that D T 1 a , D T 2 a are mutually disjoint o ver all a ∈ b Γ . Moreov er , we hav e [ S ∈C 2 D S ⊆ { 0 } × Z ∗ L ′ ∅ ∅ { 0 } × Z ∗ L ′ . (48) Finally , the third class C 3 only contains one code word giv en by U = { (1 , ( j, 0)) : j = 0 , 1 , . . . , w − 2 } ∪ { (2 , (0 , 0)) } . Observe that D U = Z ∗ w − 1 × { 0 } Z w − 1 × { 0 } Z w − 1 × { 0 } ∅ . (49) By (45), (48), (49) and aforementioned arguments, C 1 ∪ C 2 ∪ C 3 forms the desired MC-CA C. W e have the following upper bound on the value K (2 , L, w ) when the length L is a product of larger prime factors. Theorem 12. Let L ′ and w be positive inte gers. If the prime factors of L ′ ar e all lar ger than or equal to 2 w − 1 , then K (2 , ( w − 1) L ′ , w ) ≤ L ′ +  L ′ + w − 3 w − 1  . Pr oof. Let L = ( w − 1) L ′ , and let C be any code in MC-CA C (2 , L, w ) . W e first partition code words in C into two classes, C 1 and C 2 , by C 1 = { S ∈ C : e S = 1 } , C 2 = { S ∈ C : e S = 2 } . Consider the codewords in C 1 . Suppose S ∈ C 1 is exceptional. By Lemma 1(i), we have | H ( S − S ) | ≤ 2 w − 2 , which is strictly less than any factor of L ′ . Note that | H ( S − S ) | di vides L = ( w − 1) L ′ as H ( S − S ) is a subgroup of Z L due to Proposition 1(i). It must be the case that | H ( S − S ) | divides w − 1 . This is a contradiction to Lemma 1(iii). Therefore, there is no exceptional codew ord in C 1 . It follo ws that X S ∈C 1 | D S (1 , 1) | + | D S (2 , 2) | ≥ (2 w − 2) |C 1 | . (50) February 26, 2026 DRAFT 22 No w , consider the codew ords in C 2 . If { S 1 , S 2 } is an exceptional pair , by Lemma 2(i), we hav e | H ( S 1 − S 2 ) | ≤ | S 1 | + | S 2 | − 2 = w − 2 , which is strictly less than any factor of L ′ . As H ( S 1 − S 2 ) is a subgroup of Z L , it implies that | H ( S 1 − S 2 ) | divides w − 1 , which is a contradiction to Lemma 2(ii). So, { S 1 , S 2 } is not an exceptional pair . Hence, for S ∈ C 2 , one has | D S (1 , 2) | = | S 1 − S 2 | ≥ | S 1 | + | S 2 | − 1 = w − 1 . It follows that ( w − 1) L ′ ≥ X S ∈C | D S (1 , 2) | = X S ∈C 2 | D S (1 , 2) | ≥ ( w − 1) |C 2 | . (51) Follo wing the arguments abov e, if S is an exceptional code word in C 2 , it must be the case that either S 1 , S 2 , or both are exceptional. T o make a more accurate estimation of the number of dif ferences that are deriv ed from all exceptional code words, we define, for i = 1 , 2 , C ( i ) 2 ≜ { S i : S ∈ C 2 } and E ( i ) ≜ { S i ∈ C ( i ) 2 : S i is exceptional } . Note that |C (1) 2 | = |C (2) 2 | = |C 2 | and X A ∈C (1) 2 | A | + X B ∈C (1) 2 | B | = w |C 2 | . (52) Consider an element A ∈ E (1) . For notational con venience, denote by H A = H ( A − A ) . Since, by Proposition 1(i), H A contains the element 0 , so we can further denote by H ∗ A = H A \ { 0 } . By the same ar gument gi ven abo ve, | H A | must divide w − 1 . As H A is a subgroup of Z L , we hav e H A = − H A , which implies that | − A + H A | = | − ( A + H A ) | = | A + H A | . By plugging A = A and B = − A into (6), we hav e | d ( A ) | = | A + ( − A ) | ≥ | A + H A | + | − A + H A | − | H A | = 2 | A + H A | − | H A | ≥ 2 | A | − | H ∗ A | − 1 , which leads to | d ∗ ( A ) | ≥ 2 | A | − | H ∗ A | − 2 . (53) Since H A is a subgroup of Z L and | H A | di vides w − 1 , by Proposition 2, H A is a subgroup of G = { iL ′ : i = 0 , 1 , . . . , w − 2 } . Moreover , observe that H A ⊆ d ( A ) due to Proposition 1(ii). DRAFT February 26, 2026 23 Then, by the definition of an MC-CA C, H ∗ A ∩ H ∗ A ′ = ∅ for an y pair of distinct A, A ′ ∈ E (1) . This concludes that X A ∈E (1) | H ∗ A | =       ] A ∈E (1) H ∗ A       ≤ | G \ { 0 }| = w − 2 . By (53), we further hav e X A ∈E (1) | d ∗ ( A ) | ≥   X A ∈E (1) 2 | A | − 2   − ( w − 2) . Since | d ∗ ( A ) | ≥ 2 | A | − 2 if A is not exceptional, it follo ws that X A ∈C (1) 2 | d ∗ ( A ) | ≥    X A ∈C (1) 2 2 | A | − 2    − ( w − 2) . (54) Similarly , we hav e X B ∈C (2) 2 | d ∗ ( B ) | ≥    X B ∈C (2) 2 2 | B | − 2    − ( w − 2) . (55) By (50), (52), (54) and (55), 2( w − 1) L ′ − 2 = 2 | Z ∗ L | ≥ X S ∈C | D S (1 , 1) | + X S ∈C | D S (2 , 2) | = X S ∈C 1 ( | D S (1 , 1) | + | D S (2 , 2) | ) + X S ∈C 2 ( | D S (1 , 1) | + | D S (2 , 2) | ) ≥ (2 w − 2) |C 1 | + X A ∈C (1) 2 | d ∗ ( A ) | + X B ∈C (1) 2 | d ∗ ( B ) | ≥ (2 w − 2) |C 1 | + (2 w − 4) |C 2 | − 2( w − 2) , which yields ( w − 1) L ′ − 1 ≥ ( w − 1) |C 1 | + ( w − 2) |C 2 | − ( w − 2) . (56) Combining it with (51), we obtain w L ′ − 1 ≥ ( w − 1) |C | − ( w − 2) , and hence the result follo ws. W e now show that the construction described in Theorem 11 is optimal. February 26, 2026 DRAFT 24 Theorem 13. Let w be positive inte ger , and let p 1 < · · · < p n be primes such that p i − 1 is divisible by 2 w − 2 for each i . F or each 1 ≤ i ≤ n , if ther e is a code in CAC e ( p i , w ) with ( p i − 1) / (2 w − 2) codewor ds and the two conditions in (26) and (27) hold, then for any positive inte gers r 1 , . . . , r n , K (2 , L, w ) = n Y i =1 p r i i + ( Q n i =1 p r i i ) − 1 w − 1 , wher e L = ( w − 1) p r 1 1 · · · p r n n . Pr oof. The assumption that p 1 − 1 is divisible by 2 w − 2 guarantees that p 1 ≥ 2 w − 1 ≥ w . By plugging m i = ( p i − 1) / (2 w − 2) into Theorem 11, there exists a code in MC-CA C (2 , L, w ) with Q n i =1 p r i i + (( Q n i =1 p r i i ) − 1) / ( w − 1) codew ords. Let C be any code in MC-CA C (2 , L, w ) . By plugging L ′ = Q n i =1 p r i i into Theorem 12, |C | ≤ n Y i =1 p r i i + ( Q n i =1 p r i i ) − 1 w − 1 +  w − 2 w − 1  = n Y i =1 p r i i + ( Q n i =1 p r i i ) − 1 w − 1 . This completes the proof. V I . M U LT I C H A N N E L C AC S W I T H M > 2 C H A N N E L S Theorem 14. Let M , L, L ′ , and w be positive inte gers such that w > M ≥ 3 , M divides w , L = (2 w M − 1) L ′ , and all prime factors of L ′ ar e lar ger than or equal to 2 w − 1 . If ther e is a code in CA C e ( L ′ , w ) with ( L ′ − 1) / (2 w − 2) codewor ds, then ther e exists a code C ∈ MC-CA C ( M , L, w ) with |C | = M ( M − 1) L (2 w − M )( w − 1) + M ( L − 1) 2 w − 2 . Pr oof. For con venience, let t = w M . Then L = (2 t − 1) L ′ . Since ev ery prime factor p of L ′ satisfies p ≥ 2 w − 1 > 2 t − 1 , then gcd(2 t − 1 , L ′ ) = 1 , thus Z L ∼ = Z 2 t − 1 × Z L ′ . So, for the sake of con veniences, the elements in Z L can be represented as ordered pairs in Z 2 t − 1 × Z L ′ . The objecti ve code will be decomposed into two classes, C 1 and C 2 , whose constructions are gi ven belo w . Let C ′ be the code in CAC( L ′ , w ) with |C ′ | = ( L ′ − 1) / 2( w − 1) code words. For S ∈ C ′ and m ∈ I M , define a w -subset in I M × Z L by b S m ≜ { ( m, (0 , a )) ∈ I M × Z L : a ∈ S } . (57) DRAFT February 26, 2026 25 Let C 1 = [ S ∈C ′ , m ∈I M b S m . Obviously , |C 1 | = M ( L ′ − 1) / (2 w − 2) . It is easy to see that D b S m ( i, j ) =      { 0 } × d ∗ ( S ) if i = j = m, ∅ otherwise . (58) As C ′ is a CA C, it holds that D b S m , b S m ∈ C 1 , are mutually disjoint. Secondly , let Ω = { (1 , g ) ∈ Z 2 t − 1 × Z L ′ : g ∈ Z L ′ } (= { 1 } × Z L ′ ) . For a ∈ Ω , define a w -subset in I M × Z L by S a ≜ M [ m =1 { ( m, k a ) ∈ I M × Z L : k = ( m − 1) t, ( m − 1) t + 1 , . . . , mt − 1 } . (59) Let C 2 = [ a ∈ Ω S a . It is not dif ficult to see that D S a ( i, j ) =      { k a ∈ Z L : k = ± 1 , ± 2 , . . . , ± ( t − 1) } if i = j, { k a ∈ Z L : ( i − j − 1) t + 1 ≤ k ≤ ( i − j + 1) t − 1 } if i  = j. (60) For con venience, let A i,j = { x ∈ Z : ( i − j − 1) t + 1 ≤ x ≤ ( i − j + 1) t − 1 } . W e will claim that D S a ∩ D S b = ∅ for any two distinct a, b ∈ Ω . Suppose to the contrary that D S a ∩ D S b  = ∅ , where a = (1 , g ) , b = (1 , h ) for some distinct g , h ∈ Z L ′ . There are two cases according to (59). Case 1. If D S a ( i, i ) ∩ D S b ( i, i )  = ∅ for some i , then there exist k , k ′ ∈ {± 1 , ± 2 , . . . , ± ( t − 1) } such that k (1 , g ) = k ′ (1 , h ) in Z 2 t − 1 × Z L ′ . This implies that k = k ′ and thus k ( g − h ) = 0 (mod L ′ ). Since all prime f actors of L ′ are lar ger than or equal to 2 w − 1 , which is strictly larger than 2 t − 1 , we hav e gcd( k , L ′ ) = 1 . This further implies that g = h , which is a contradiction. Case 2. If D S a ( i, j ) ∩ D S b ( i, j )  = ∅ for some i  = j , then there exist integers k , k ′ ∈ A i,j such that k (1 , g ) = k ′ (1 , h ) in Z 2 t − 1 × Z L ′ . Observe that A i,j is consist of consecuti ve 2 t − 1 integers. Then, we get k = k ′ , and thus k ( g − h ) = 0 (mod L ′ ). Moreov er , since i, j are distinct elements in I M , we have − w + 1 ≤ k ≤ w + 1 and k  = 0 . Again, since all prime factors of L ′ February 26, 2026 DRAFT 26 are larger than or equal to 2 w − 1 , one has gcd( k , L ′ ) = 1 . Hence, it follows that g = h , and a contradiction occurs. Furthermore, by (58) and (60), it is obvious that D S ∩ D S ′ = ∅ whene ver S ∈ C 1 and S ′ ∈ C 2 . It concludes that C = C 1 ⊎ C 2 is a code in MC-CA C ( M , L, w ) . Finally , as L ′ = M L/ (2 w − M ) , the code size is |C | = |C 1 | + |C 2 | = M ( L ′ − 1) 2 w − 2 + L ′ = M + 2 w − 2 2 w − 2 · M L 2 w − M − M 2 w − 2 = M ( M − 1) L ( w − 1)(2 w − M ) + M ( L − 1) 2 w − 2 , as desired. For any positi ve integer n , let τ ( n ) , called the number-of-divisor function , denote the number of positi ve di visors of n . Theorem 15. Let M , L, L ′ and w be positive inte gers with w > M ≥ 3 such that M divides w , L = (2 w M − 1) L ′ , and all prime factors of L ′ ar e lar ger than or equal to 2 w − 1 . Then, K ( M , L, w ) ≤ M ( M − 1)( L + ( τ (2 w M − 1) − 1)( L − L ′ )) (2 w − M )( w − 1) + M ( L − 1) + 2( w − M ) 2 w − 2 . Pr oof. Let C be any code in MC-CA C ( M , L, w ) . W e adopt the notation from the proof of Theorem 4 for con venience. Let C 1 = { S ∈ C : e S = 1 } , C 2 = { S ∈ C : e S = M } , and C 3 = { S ∈ C : 2 ≤ e S ≤ M − 1 } . For i ∈ I M , let C ( i ) ≜ { S i : S ∈ C } and E ( i ) ≜ { A ∈ C ( i ) : A is exceptional } denote the collections of all subsets and all e xceptional subsets associated with the i -th channel, respecti vely . For A ∈ E ( i ) , let H A = H ( A − A ) for notational con venience. For any fixed index i ∈ I M , by the same arguments used to obtain (54), we deriv e X A ∈E ( i ) | d ∗ ( A ) | ≥   X A ∈E ( i ) 2 | A | − 2   −  2 w M − 2  . DRAFT February 26, 2026 27 Note that | d ∗ ( A ) | ≥ 2 | A | − 2 for all A ∈ C ( i ) \ E ( i ) . Therefore, X A ∈C ( i ) | d ∗ ( A ) | ≥   X A ∈C ( i ) 2 | A | − 2   − 2  w M − 1  . (61) Summing (61) ov er all i ∈ I M , by the disjoint-dif ference-array property , yields M ( L − 1) ≥ 2( w − 1) |C 1 | + 2( w − M ) |C 2 | + 2 X S ∈C 3 ( w − e S ) − 2 M  w M − 1  . (62) No w , we consider D S ( i, j ) ( = S i − S j ) for distinct indices i, j ∈ I M . Note that S i − S j = ∅ for S ∈ C 1 . For S ∈ C 2 ∪ C 3 , denote by H ( i,j ) S = H ( S i − S j ) for notational con venience. By plugging A = S i and B = − S j into (7), we hav e | S i − S j | ≥ | S i | + | S j | − | H ( i,j ) S | . (63) For two distinct channel indices i, j ∈ I M , let E ( i,j ) ≜ { S ∈ C 2 ∪ C 3 : { S i , S j } is an exceptional pair } denote the collections of all exceptional pairs associated with the i - and j -th channels. Note that | S i − S j | ≥ | S i | + | S j | − 1 when { S i , S j } is not an exceptional pair . By (63) and the disjoint-dif ference-array property , we have L = | Z L | ≥ X S ∈C 2 ∪C 3 | S i − S j | ≥ X S ∈C 2 ∪C 3 ( | S i | + | S j | − 1) − X S ∈E ( i,j )  | H ( i,j ) S | − 1  . By the same arguments used to obtain (18), we can sum the abov e inequality ov er all distinct indices i, j ∈ I M to deri ve M ( M − 1) L ≥ ( M − 1)(2 w − M ) |C 2 | + X S ∈C 3 ( e S − 1)(2 w − e S ) − X i,j ∈I M ,i  = j X S ∈E ( i,j )  | H ( i,j ) S | − 1  . (64) By multiplying the inequality in (62) by (2 w − M ) / 2 and combining the result with (64), we get ( w − 1)(2 w − M )( |C 1 | + |C 2 | ) + X S ∈C 3 ( e S − 1)(2 w − e S ) ≤ M ( M − 1) L + X i,j ∈I M ,i  = j X S ∈E ( i,j )  | H ( i,j ) S | − 1  + M ( L − 1)(2 w − M ) 2 + (2 w − M )( w − M ) February 26, 2026 DRAFT 28 Note that we ha ve dealt with the summation P S ∈C 3 ( e S − 1)(2 w − e S ) in the proof of Theorem 4. By (21), the abov e inequality can be simplified to |C | ≤ M ( M − 1) L + P i,j ∈I M ,i  = j P S ∈E ( i,j )  | H ( i,j ) S | − 1  (2 w − M )( w − 1) + M ( L − 1) + 2( w − M ) 2 w − 2 . (65) For S ∈ E ( i,j ) , by Lemma 2(i), | H ( i,j ) S | ≥ 2 . It follows from Proposition 1(iii) that | H ( i,j ) S | di vides | S i − S j | . Note that | H ( i,j ) S | di vides L since H ( i,j ) S is a subgroup of Z L . For a gi ven subset H of Z L , define δ ( H ) as the collection of elements S ∈ E ( i,j ) such that H ( i,j ) S = H . Since, for two distinct S, S ′ ∈ δ ( H ) , ( S i − S j ) ∩ ( S ′ i − S ′ j ) = ∅ due to the disjoint-dif ference-array property , we hav e X S ∈ δ ( H ) | S i − S j | ≤ L. This implies that | δ ( H ) | ≤ L | H | . In other words, there are at most L/ | H | exceptional pairs { S i , S j } over all E ( i,j ) such that their set of stabilizers is H . By Lemma 2(i) again, | H ( i,j ) S | ≤ | S i | + | S j | − 2 ≤ w − 2 , which is strictly less than any prime factor of L ′ . By Lagrange’ s theorem, the value | H ( i,j ) S | divides (2 w M − 1) . Moreov er , | H ( i,j ) S | ≥ 2 . That means, there are at most τ (2 w M − 1) − 1 candidates for the set of stabilizers of an exceptional pair associated with E ( i,j ) . Hence, we conclude that X S ∈E ( i,j )  | H ( i,j ) S | − 1  ≤  τ (2 w M − 1) − 1  · L | H | ( | H | − 1) ≤  τ (2 w M − 1) − 1  ( L − L ′ ) , where the second inequality is due to | H | being a factor of (2 w M − 1) . Finally , we get X i,j ∈I M ,i  = j X S ∈E ( i,j )  | H ( i,j ) S | − 1  ≤ M ( M − 1)  τ (2 w M − 1) − 1  ( L − L ′ ) . The result follo ws by (65). V I I . T W O E X T E N S I O N S A. At Most One-P ack et P er T ime Slot MC-CA Cs A code C ∈ MC-CA C ( M , L, w ) is said to ha ve at most one-pac ket per time slot (AM-OPPTS) if, for all codew ord S ∈ C , the element “one” appears at most once in each column [4]. That is, DRAFT February 26, 2026 29 |{ m : ( m, t ) ∈ S }| ≤ 1 for each t ∈ Z L . Obviously , the AM-OPPTS property is meaningless if L < w . Let AM-OPPTS MC-CAC ( M , L, w ) be the collection of codes in MC-CA C ( M , L, w ) with AM-OPPTS property . The maximum size of a code in AM-OPPTS MC-CA C ( M , L, w ) is denoted by A ( M , L, w ) , i.e., A ( M , L, w ) ≜ max {|C | : C ∈ AM-OPPTS MC-CA C ( M , L, w ) } . By definitions, we hav e A ( M , L, w ) ≤ K ( M , L, w ) for any integers M , L and w . The follo wing upper bound of A ( M , L, w ) is due to [4, Corollaries 3.6, 3.7]. Theorem 16 ([4]) . Let M , L and w be positive inte gers. If the prime factors of L ar e all lar ger than or equal to 2 w − 1 , we have A ( M , L, w ) ≤ M ( M − 1)( L − 1) w ( w − 1) + M ( L − 1) 2 w − 2 . (66) The upper bounds in Theorem 16 can be impro ved in the case where M < w , which is shown in the follo wing theorem. Theorem 17. Let M , L and w be positive inte gers with M < w . If the prime factors of L ar e all lar ger than or equal to 2 w − 1 , we have A ( M , L, w ) ≤ M ( M − 1)( L − 1) (2 w − M )( w − 1) + M ( L − 1) 2 w − 2 . (67) Pr oof. The characteristic of a code C in AM-OPPTS MC-CA C ( M , L, w ) is that, 0 / ∈ D S ( i, j ) for any codeword S in volv ed and any distinct indices i, j ∈ I M . Therefore, the inequality in (67) can be obtained in the same arguments giv en in the proof of Theorem 4, b ut just modifying (17) to be X S ∈C 2 ∪C 3 | D S ( i, j ) | ≤ | Z ∗ L | = L − 1 , which changes the last term M ( M − 1) L on the right-hand-side in (19) to M ( M − 1)( L − 1) . Hence, the result follo ws. W e remark that an alternate proof of Theorem 16 can be obtained by means of the ar guments in Appendix A. Based on the constructions de veloped in Sections V and VI, we obtain the following results. February 26, 2026 DRAFT 30 Theorem 18. Let p 1 < · · · < p n be primes such that p i − 1 is divisible by 2 w − 2 for eac h i . F or each 1 ≤ i ≤ n , if ther e is a code in CAC e ( p i , w ) with ( p i − 1) / (2 w − 2) code wor ds and the two conditions in (26) and (27) hold, then for any positive inte gers r 1 , . . . , r n , n Y i =1 p r i i + ( Q n i =1 p r i i ) − 1 w − 1 − 1 ≤ A (2 , L, w ) ≤ n Y i =1 p r i i + ( Q n i =1 p r i i ) − 1 w − 1 , wher e L = ( w − 1) p r i 1 · · · p r n n . Pr oof. The lower bound can be deriv ed from the construction in the proof of Theorem 11 by ignoring the last codew ord, say the unique one in C 3 . The upper bound is a consequence of Theorem 13 because A (2 , L, w ) ≤ K (2 , L, w ) . Theorem 19. Let M , L, L ′ , and w be positive inte gers such that w > M ≥ 3 , M divides w , L = (2 w M − 1) L ′ , and all prime factors of L ′ ar e lar ger than or equal to 2 w − 1 . If ther e is a code in CA C e ( L ′ , w ) with ( L ′ − 1) / (2 w − 2) codewor ds, then M ( M − 1) L (2 w − M )( w − 1) + M ( L − 1) 2 w − 2 ≤ A ( M , L, w ) ≤ M ( M − 1)( L + ( τ (2 w M − 1) − 1)( L − L ′ )) (2 w − M )( w − 1) + M ( L − 1) + 2( w − M ) 2 w − 2 . Pr oof. In the proof of Theorem 14, it is easy to see from (60) that 0 / ∈ D S a ( i, j ) for i  = j . This implies that the code obtained in Theorem 14 is has the AM-OPPTS property . The upper bound is a consequence of Theorem 15 because A ( M , L, w ) ≤ K ( M , L, w ) . B. Mixed-W eight Codes [5] introduced mix ed-weight CA Cs for the purpose of increasing the throughput and reducing the access delay of some potential users with higher priority . This section will provide some constructions of mixed-weight CA Cs, which can be seen as generalizations of Theorem 6 and Theorem 9. Furthermore, the concept of mix ed-weight will also be e xtended to a multichannel version by relaxing the identical-weight constraint in prior studies of MC-CA Cs. Definition 5. Let L be a positi ve inte ger and W be a set of positi ve inte gers. A mixed-weight CA C C of length L with weight-set W is a collection of subsets of Z L such that (i) each subset is of size in W ; and (ii) C satisfies the disjoint-difference-set property as shown in (1). DRAFT February 26, 2026 31 W e can use the method giv en in [5, Theorem 16] to construct a mixed-weight CA C of length ( w − 1) p r 1 1 · · · p r n n with weight-set W = { w − 1 , w , w ∗ i 1 , w ∗ i 2 , . . . , w ∗ i t } provided a code in CA C e ( p r i j i j , w ∗ i j ) exists for some i 1 , i 2 , . . . , i k . Theorem 20. Let w, r 1 , r 2 , . . . , r n be positive inte ger s and p 1 < p 2 < · · · < p n be primes such that p 1 ≥ w . Suppose , for each 1 ≤ i ≤ n , the two conditions given in (26) and (27) hold and ther e exists a code A i ∈ CAC( p r i i , w ∗ i ) that contains m i equi-differ ence codewor ds, where w ∗ i is an arbitrary positive inte ger and m i is not necessarily non-zer o. Then ther e exists a code C ∈ CA C( L, W ) with |C | = p r 1 1 · · · p r n n − 1 2 + 1 + n X i =1 m i , wher e L = ( w − 1) p r 1 1 · · · p r n n and W = { w − 1 , w , w ∗ 1 , w ∗ 2 , . . . , w ∗ n } . In particular , if all code wor ds in eac h A i ar e non-e xceptional, then C contains 1 + P n i =1 m i ( w ∗ i − 1) codewor ds with weight w − 1 , p r 1 1 ··· p r n n 2 − P n i =1 m i ( w ∗ i − 1) code words with weight w , and m i codewor ds with weight w ∗ i for each i . Theorem 21. Let p 1 < · · · < p n be primes and w 1 , . . . , w n be positive inte gers (not necessarily distinct) such that p i ≥ 2 w i − 1 for any i . If ther e is a code in CA C e ( p i , w i ) with m i codewor ds for each i , then for any positive inte gers r 1 , . . . , r n , ther e e xists a mixed-weight code C ∈ CA C( L, W ) with |C | = n X i =1 m i ( p r i i − 1) p i − 1 n Y j = i +1 p r j j , wher e L = p r 1 1 · · · p r n n and W = { w 1 , . . . , w n } . In particular , C contains m i ( p r i i − 1) p i − 1 n Y j = i +1 p r j j codewor ds with weight w i , for i = 1 , 2 , . . . , n . Pr oof. The proof can be done with the same ar guments in the proof of Theorem 9 by adjusting Γ i being the set of m i generators of the gi ven code in CA C e ( p i , w i ) and replacing the definition of S a in (37) with S a = { j a ∈ Z p r 1 1 × · · · × Z p r n n : j = 0 , 1 , 2 , . . . , w i − 1 } . February 26, 2026 DRAFT 32 W e now extend the concept of mixed-weight to multichannel CA Cs. Definition 6. Let M , L be two positiv e inte gers and W be a set of positive integers. A mixed- weight MC-CAC C of length L with weight-set W in M channels is a collection of subsets of I M × Z L such that (i) each subset is of size in W ; and (ii) C satisfies the disjoint-dif ference-array property as sho wn in (3). V I I I . C O N C L U D I N G R E M A R K S It has been sho wn in [1, Theorem 1] that lim sup L →∞ K ( L, w ) L = 1 2 w − 2 . (68) When considering multichannel CA Cs, if M = 2 , it re veals from Theorem 13 that K (2 , L, w ) L ∼ 1 w − 1 + 1 ( w − 1) 2 as L → ∞ (69) under some specific parameter settings. For general M ≥ 3 , when w > M and under other specific parameter settings, we can see from Theorem 14 that the lo wer bound of K ( M , L, w ) /L approaches M ( M − 1) (2 w − M )( w − 1) + M 2 w − 2 (70) as L goes to infinity . Ho wev er , based on Theorem 15, our deri ved upper bound of K ( M , L, w ) /L approaches M ( M − 1)  1 + ( τ ( m ) − 1)(1 − 1 m )  (2 w − M )( w − 1) + M 2 w − 2 (71) as L goes to infinity , where m = 2 w M − 1 . One can see that the asymptotic value in (70) reduces to (68) and (69) when M = 1 and 2 , respecti vely . Moti vated by (68), we hav e the following conjecture. Conjecture 1. F or all w ≥ M , we have lim sup L →∞ K ( M , L, w ) L = M ( M − 1) (2 w − M )( w − 1) + M 2 w − 2 . This conjecture remains an open problem for future in vestigation. DRAFT February 26, 2026 33 A P P E N D I X A A N E W P R O O F O F T H E O R E M 3 Let C be any code in MC-CA C ( M , L, w ) . Similar to the proof of Theorem 4, all code words are non-exceptional and partitioned into the three classes C 1 = { S ∈ C : e S = 1 } , C 2 = { S ∈ C : e S = w } , and C 3 = { S ∈ C : 2 ≤ e S ≤ w − 1 } . By the same argument in the proof of Theorem 4, we hav e ( w − 1) |C 1 | + X S ∈C 3 ( w − e S ) ≤ M ( L − 1) 2 , (72) and w ( w − 1) |C 2 | + X S ∈C 3 ( e S − 1)(2 w − e S ) ≤ M ( M − 1) L, (73) which are analogous to (16) and (18), respectiv ely . Multiplying the inequality in (72) by w and combining the result with (73), we obtain w ( w − 1)( |C 1 | + |C 2 | ) + X S ∈C 3  − e 2 S + ( w + 1) e S + w 2 − 2 w  ≤ w M ( L − 1) 2 + M ( M − 1) L. (74) Define a function f ( x ) = − x 2 + ( w + 1) x + w 2 − 2 w on the interv al [2 , w − 1] . It is not hard to see f ( x ) has the minimum value w 2 − 2 at x = 2 or x = w − 1 , which leads to f ( x ) ≥ w 2 − 2 ≥ w ( w − 1) (75) because of w ≥ 2 . Hence, it follo ws from (74) that w ( w − 1) |C | ≤ w M ( L − 1) 2 + M ( M − 1) L, (76) as desired. Note that in (75), f ( x ) > w ( w − 1) if w ≥ 3 , leading to a strict inequality in (76). 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