You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


I. Field of the Invention
This invention relates to the field of communications systems and, in particular, to the transmission of spread coded message signals within communications systems.
II. Description of the Prior Art
It is well known in the art of communications to mix message signals to be transmitted with spreading code vectors. This permits the message signals to be combined, transmitted, and separated from each other after transmission. The most useful feature of a set of code vectors suitable for this purpose is that the spreading code vectors are mutually orthogonal. This permits a theoretical interference of zero between the message signals. The code vectors most commonly used for this purpose are Walsh code vectors.
The total number of binary code vectors having a length n is 2n. However, of the total number of binary vectors 2n within the total vector space, only n are mutually orthogonal. For example, when n=8 there are 256 different binary vectors. Only 8 of the 256 vectors are mutually orthogonal. Therefore, in a system wherein n=8 usually only 8 message signals can be combined and separated in this manner and only 8 users can be supported simultaneously. Likewise, if n=128 then 128 users can be supported simultaneously. Some of the vectors can be idle some of the time, thereby permitting more than n users to be serviced. However, the size of the code vectors still places a limitation on the size of the communications system.
A set W of code vectors w meeting the orthogonality requirement for a theoretical interference of zero can be represented as follows:                               w          1                =                  [                                    w                              1                ,                1                                      ⁢                          w                              1                ,                2                                      ⁢                                          …                ⁢                w                                            1                ,                n                                              ]                                                  w          2                =                  [                                    w                              2                ,                1                                      ⁢                          w                              2                ,                2                                      ⁢                                          …                ⁢                w                                            2                ,                n                                              ]                                ⋮                                    w          n                =                  [                                    w                              n                ,                1                                      ⁢                          w                              n                ,                2                                      ⁢                                          …                ⁢                w                                            n                ,                n                                              ]                    
wherein each vector wi is a column vector using a 0/1 alphabet or, equivalently, a xe2x88x921/+1 alphabet. Hereinbelow, a set of code vectors using the 0/1 alphabet is expressed as Wb,n and a set using the xe2x88x921/+1 alphabet is expressed as Wn.
Since all vectors w in the set W are orthogonal to each other, the dot product of any two vectors in the set must be zero. This can be represented as:
(wx, wy)=0 
where x and y can have any values between 1 and n, xxe2x89xa0y and (wx, wy) is equal to:             ∑              i        =        1            n        ⁢          w              x        ,        i              ,            w              y        ,        i              .  
Equivalently, the above can be expressed as the following matrix product:
wxTwy=0. 
Also:
wxTwx=n. 
Representing the ith data symbol to be transmitted as di and the total number of transmit signals as k, the total transmission signal S transmitted by a base station to a mobile station is:   S  =            ∑              i        =        1            k        ⁢                  d        i            ⁢              w        i            
The mobile station receives the total transmission signal S and tries to eliminate all message signals except its own.
In order to eliminate the other messages the mobile station can multiply signal S by the transpose of its own Walsh code vector. An example wherein i=1 is as follows:                                           w            1            T                    ⁢          S                =                              w            1            T                    ⁢                                    ∑                              i                =                1                            k                        ⁢                                          d                i                            ⁢                              w                i                                                                            =                              w            1            T                    ⁡                      (                                                            d                  1                                ⁢                                                      w                    _                                    1                                            +                                                ∑                                      i                    =                    2                                    k                                ⁢                                                      d                    i                                    ⁢                                                            w                      _                                        i                                                                        )                              
wherein the first term on the right side represents the wanted signal. The second term in the right side represents the interference from all of the remaining message signals mixed with their individual Walsh codes. Solving this equation yields:
w1TS=nd1+0. 
Thus, the separation of the transmitted message signals at the receiver depends on a zero correlation between the wanted signal and all of the other message signals.
In order to utilize communications systems as effectively as possible it is desirable to simultaneously transmit and separate as many message signals as possible. However, it is only possible to mix n message signals and separate them with zero interference because only n orthogonal vectors are available, as previously described. To overcome this limitation it is known to use quasi orthogonal functions. Quasi orthogonal vectors are vectors that are in addition to the n orthogonal vectors. Quasi orthogonal vectors have been selected from the remaining code vectors in the total binary 2n vector space in order to provide as little interference as possible. Specifically, quasi orthogonal vectors are selected to provide a level of interference that is within acceptable limits, even though the level of interference is not zero.
In order to select quasi orthogonal vectors a computer search can be performed within the total 2n vector space for binary (+1/xe2x88x921 alphabet) masks. The masks can be applied to the orthogonal vectors to form a new set of vectors that are quasi orthogonal vectors. Applying a total of M masks to a set of Walsh code vectors wn, the number of quasi orthogonal functions produced is: (M+1)n. Applying a mask m to a code vector wxcex5Wn includes a component by component multiplication of the mask m and the orthogonal code vector w to give the new code vector:
wm=wxc2x7m 
The interference resulting from the use of the new code vectors can be tested and the code vectors that provide the lowest correlation can be selected to provide a set of quasi orthogonal vectors. A plurality of such masking functions can be found in order to provide a plurality of sets of quasi orthogonal vectors from a single set of orthogonal vectors. In order to permit message signals mixed with the quasi orthogonal vectors found by the computer search to be separated from each other, the quasi orthogonal vectors should be mutually orthogonal with respect to each other. There is a non-zero correlation between at least one code vector in the orthogonal set and one vector in the quasi orthogonal set.
Representing the quasi orthogonal vectors as v it can be shown that:             1      n        ⁢                  ∑                  j          +          1                n            ⁢              (                              (                          v              ,                              w                j                                      )                    2                )              =      1    n  
The goal in picking quasi orthogonal vectors v is to pick the vectors such that   1  ≤  i  ≤      n    ⁢                  {                  "LeftBracketingBar"                      (                                          v                _                            ,                                                w                  _                                i                                      )                    "RightBracketingBar"                }            max      
is as small as possible.
Since their correlation is a useful measure of the amount of separation between vectors, the normalized correlation between two code vectors x and y can be defined as:       (                  x        _            ,              y        _              )    =            1      n        ⁢                  ∑                  i          =          1                n            ⁢                        x          1                ⁢                  y          i          *                    
The correlation between two orthogonal vectors is zero. A lower absolute value of correlation results in better separation between message signals mixed with the orthogonal vectors and the ones mixed with quasi orthogonal vectors. Better signal separation results in lower interference between the signals at the time of decoding. Better signal separation results in lower interference between the signals at the time of decoding.
The mean square correlation between orthogonal vectors and their corresponding quasi orthogonal vectors where n is a power of two is 1/n. The lower bound on the absolute value of correlation can be shown to have the value, 1/{square root over (n)}. This quantity is referred to as the Holtzman lower bound. Masks have been found that meet the lower bound for cases wherein n is an even power of two. However, in cases where n is an odd power of two this bound has not been met with an equality. The lowest correlation found in the latter case is {square root over (2)}/{square root over (n)}. Therefore, the interference of the best quasi orthogonal vectors found in the odd power of two case using the computer search technique is {square root over (2)} times the theoretical limit.
Thus it desirable to find additional quasi orthogonal vectors having lower correlation with the orthogonal vectors for the case wherein n is an odd power of two, in order to expand the capacity of communications systems while maintaining acceptably low amounts of interference.
A transmission method in a communications system has an orthogonal vector and a quasi orthogonal masking function for obtaining a quasi orthogonal vector from the orthogonal vector. Message signals are transmitted according to the quasi orthogonal vector. The method includes receiving the quasi orthogonal masking function and permuting the quasi orthogonal masking function to provide a further quasi orthogonal masking function. The further quasi orthogonal masking function is applied to the orthogonal vector to provide a further quasi orthogonal vector. The further quasi orthogonal vector is applied to the message signal to provide an encoded message signal for transmitting the encoded message signal within the communications system.